Apparatus for magnetic and electrostatic confinement of plasma

ABSTRACT

An apparatus and method for containing plasma and forming a Field Reversed Configuration (FRC) magnetic topology are described in which plasma ions are contained magnetically in stable, non-adiabatic orbits in the FRC. Further, the electrons are contained electrostatically in a deep energy well, created by tuning an externally applied magnetic field. The simultaneous electrostatic confinement of electrons and magnetic confinement of ions avoids anomalous transport and facilitates classical containment of both electrons and ions. In this configuration, ions and electrons may have adequate density and temperature so that upon collisions ions are fused together by nuclear force, thus releasing fusion energy. Moreover, the fusion fuel plasmas that can be used with the present confinement system and method are not limited to neutronic fuels only, but also advantageously include advanced fuels.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. Ser. No. 11/173,204 filedJul. 1, 2005, now U.S. Pat. No. 7,129,656, which is a divisional of U.S.Ser. No. 10/328,703, filed Dec. 23, 2002, now U.S. Pat. No. 7,026,763,which is a continuation of U.S. Ser. No. 10/066,424, filed Jan. 31,2002, now U.S. Pat. No. 6,664,740, which claims the benefit ofprovisional U.S. application Ser. No. 60/266,074, filed Feb. 1, 2001 andprovisional U.S. application Ser. No. 60/297,086, filed on Jun. 8, 2001,which applications are fully incorporated herein by reference.

This invention was made with Government support under Contract No.N00014-99-1-0857, awarded by the Office of Naval Research. Somebackground research was supported by the U.S. Department of Energy for1992 to 1993. The Government has certain rights in this invention.

FIELD OF THE INVENTION

The invention relates generally to the field of plasma physics, and, inparticular, to methods and apparati for confining plasma. Plasmaconfinement is particularly of interest for the purpose of enabling anuclear fusion reaction.

BACKGROUND OF THE INVENTION

Fusion is the process by which two light nuclei combine to form aheavier one. The fusion process releases a tremendous amount of energyin the form of fast moving particles. Because atomic nuclei arepositively charged—due to the protons contained therein—there is arepulsive electrostatic, or Coulomb, force between them. For two nucleito fuse, this repulsive barrier must be overcome, which occurs when twonuclei are brought close enough together where the short-range nuclearforces become strong enough to overcome the Coulomb force and fuse thenuclei. The energy necessary for the nuclei to overcome the Coulombbarrier is provided by their thermal energies, which must be very high.For example, the fusion rate can be appreciable if the temperature is atleast of the order of 10⁴ eV—corresponding roughly to 100 milliondegrees Kelvin. The rate of a fusion reaction is a function of thetemperature, and it is characterized by a quantity called reactivity.The reactivity of a D—T reaction, for example, has a broad peak between30 keV and 100 keV.

Typical fusion reactions include:D+D→He³(0.8 MeV)+n(2.5 MeV),D+T→α(3.6 MeV)+n(14.1 MeV),D+He³→α(3.7 MeV)+p(14.7 MeV), andp+B¹¹→3α(8.7 MeV),where D indicates deuterium, T indicates tritium, α indicates a heliumnucleus, n indicates a neutron, p indicates a proton, He indicateshelium, and B¹¹ indicates Boron-11. The numbers in parentheses in eachequation indicate the kinetic energy of the fusion products.

The first two reactions listed above—the D—D and D—T reactions—areneutronic, which means that most of the energy of their fusion productsis carried by fast neutrons. The disadvantages of neutronic reactionsare that (1) the flux of fast neutrons creates many problems, includingstructural damage of the reactor walls and high levels of radioactivityfor most construction materials; and (2) the energy of fast neutrons iscollected by converting their thermal energy to electric energy, whichis very inefficient (less than 30%). The advantages of neutronicreactions are that (1) their reactivity peaks at a relatively lowtemperature; and (2) their losses due to radiation are relatively lowbecause the atomic numbers of deuterium and tritium are 1.

The reactants in the other two equations—D—He³ and p-B¹¹—are calledadvanced fuels. Instead of producing fast neutrons, as in the neutronicreactions, their fusion products are charged particles. One advantage ofthe advanced fuels is that they create much fewer neutrons and thereforesuffer less from the disadvantages associated with them. In the case ofD—He³, some fast neutrons are produced by secondary reactions, but theseneutrons account for only about 10 percent of the energy of the fusionproducts. The p-B¹¹ reaction is free of fast neutrons, although it doesproduce some slow neutrons that result from secondary reactions butcreate much fewer problems. Another advantage of the advanced fuels isthat the energy of their fusion products can be collected with a highefficiency, up to 90 percent. In a direct energy conversion process,their charged fusion products can be slowed down and their kineticenergy converted directly to electricity.

The advanced fuels have disadvantages, too. For example, the atomicnumbers of the advanced fuels are higher (2 for He³ and 5 for B¹¹).Therefore, their radiation losses are greater than in the neutronicreactions. Also, it is much more difficult to cause the advanced fuelsto fuse. Their peak reactivities occur at much higher temperatures anddo not reach as high as the reactivity for D—T. Causing a fusionreaction with the advanced fuels thus requires that they be brought to ahigher energy state where their reactivity is significant. Accordingly,the advanced fuels must be contained for a longer time period whereinthey can be brought to appropriate fusion conditions.

The containment time for a plasma is Δt=r²/D, where r is a minimumplasma dimension and D is a diffusion coefficient. The classical valueof the diffusion coefficient is D_(c)=a_(i) ²/τ_(ie), where a_(i) is theion gyroradius and τ_(ie) is the ion-electron collision time. Diffusionaccording to the classical diffusion coefficient is called classicaltransport. The Bohm diffusion coefficient, attributed toshort-wavelength instabilities, is D_(B)=( 1/16)a_(i) ²Ω_(i), whereΩ_(i) is the ion gyrofrequency. Diffusion according to this relationshipis called anomalous transport. For fusion conditions, D_(B)/D_(C)=(1/16)Ω_(i)τ_(ie)≅10⁸, anomalous transport results in a much shortercontainment time than does classical transport. This relation determineshow large a plasma must be in a fusion reactor, by the requirement thatthe containment time for a given amount of plasma must be longer thanthe time for the plasma to have a nuclear fusion reaction. Therefore,classical transport condition is more desirable in a fusion reactor,allowing for smaller initial plasmas.

In early experiments with toroidal confinement of plasma, a containmenttime of Δt≅r²/D_(B) was observed. Progress in the last 40 years hasincreased the containment time to Δt≅1000 r²/D_(B). One existing fusionreactor concept is the Tokamak. The magnetic field of a Tokamak 68 and atypical particle orbit 66 are illustrated in FIG. 5. For the past 30years, fusion efforts have been focussed on the Tokamak reactor using aD—T fuel. These efforts have culminated in the InternationalThermonuclear Experimental Reactor (ITER), illustrated in FIG. 7. Recentexperiments with Tokamaks suggest that classical transport, Δt≅r²/D_(c),is possible, in which case the minimum plasma dimension can be reducedfrom meters to centimeters. These experiments involved the injection ofenergetic beams (50 to 100 keV), to heat the plasma to temperatures of10 to 30 keV. See W. Heidbrink & G. J. Sadler, 34 Nuclear Fusion 535(1994). The energetic beam ions in these experiments were observed toslow down and diffuse classically while the thermal plasma continued todiffuse anomalously fast. The reason for this is that the energetic beamions have a large gyroradius and, as such, are insensitive tofluctuations with wavelengths shorter than the ion gyroradius (λ<a_(i)).The short-wavelength fluctuations tend to average over a cycle and thuscancel. Electrons, however, have a much smaller gyroradius, so theyrespond to the fluctuations and transport anomalously.

Because of anomalous transport, the minimum dimension of the plasma mustbe at least 2.8 meters. Due to this dimension, the ITER was created 30meters high and 30 meters in diameter. This is the smallest D—TTokamak-type reactor that is feasible. For advanced fuels, such as D—He³and p-B¹¹, the Tokamak-type reactor would have to be much larger becausethe time for a fuel ion to have a nuclear reaction is much longer. ATokamak reactor using D—T fuel has the additional problem that most ofthe energy of the fusion products energy is carried by 14 MeV neutrons,which cause radiation damage and induce reactivity in almost allconstruction materials due to the neutron flux. In addition, theconversion of their energy into electricity must be by a thermalprocess, which is not more than 30% efficient.

Another proposed reactor configuration is a colliding beam reactor. In acolliding beam reactor, a background plasma is bombarded by beams ofions. The beams comprise ions with an energy that is much larger thanthe thermal plasma. Producing useful fusion reactions in this type ofreactor has been infeasible because the background plasma slows down theion beams. Various proposals have been made to reduce this problem andmaximize the number of nuclear reactions.

For example, U.S. Pat. No. 4,065,351 to Jassby et al. discloses a methodof producing counterstreaming colliding beams of deuterons and tritonsin a toroidal confinement system. In U.S. Pat. No. 4,057,462 to Jassbyet al., electromagnetic energy is injected to counteract the effects ofbulk equilibrium plasma drag on one of the ion species. The toroidalconfinement system is identified as a Tokamak. In U.S. Pat. No.4,894,199 to Rostoker, beams of deuterium and tritium are injected andtrapped with the same average velocity in a Tokamak, mirror, or fieldreversed configuration. There is a low density cool background plasmafor the sole purpose of trapping the beams. The beams react because theyhave a high temperature, and slowing down is mainly caused by electronsthat accompany the injected ions. The electrons are heated by the ionsin which case the slowing down is minimal.

In none of these devices, however, does an equilibrium electric fieldplay any part. Further, there is no attempt to reduce, or even consider,anomalous transport.

Other patents consider electrostatic confinement of ions and, in somecases, magnetic confinement of electrons. These include U.S. Pat. No.3,258,402 to Farnsworth and U.S. Pat. No. 3,386,883 to Farnsworth, whichdisclose electrostatic confinement of ions and inertial confinement ofelectrons; U.S. Pat. No. 3,530,036 to Hirsch et al. and U.S. Pat. No.3,530,497 to Hirsch et al. are similar to Farnsworth; U.S. Pat. No.4,233,537 to Limpaecher, which discloses electrostatic confinement ofions and magnetic confinement of electrons with multipole cuspreflecting walls; and U.S. Pat. No. 4,826,646 to Bussard, which issimilar to Limpaecher and involves point cusps. None of these patentsconsider electrostatic confinement of electrons and magnetic confinementof ions. Although there have been many research projects onelectrostatic confinement of ions, none of them have succeeded inestablishing the required electrostatic fields when the ions have therequired density for a fusion reactor. Lastly, none of the patents citedabove discuss a field reversed configuration magnetic topology.

The field reversed configuration (FRC) was discovered accidentallyaround 1960 at the Naval Research Laboratory during theta pinchexperiments. A typical FRC topology, wherein the internal magnetic fieldreverses direction, is illustrated in FIG. 8 and FIG. 10, and particleorbits in a FRC are shown in FIG. 11 and FIG. 14. Regarding the FRC,many research programs have been supported in the United States andJapan. There is a comprehensive review paper on the theory andexperiments of FRC research from 1960-1988. See M. Tuszewski, 28 NuclearFusion 2033, (1988). A white paper on FRC development describes theresearch in 1996 and recommendations for future research. See L. C.Steinhauer et al., 30 Fusion Technology 116 (1996). To this date, in FRCexperiments the FRC has been formed with the theta pinch method. Aconsequence of this formation method is that the ions and electrons eachcarry half the current, which results in a negligible electrostaticfield in the plasma and no electrostatic confinement. The ions andelectrons in these FRCs were contained magnetically. In almost all FRCexperiments, anomalous transport has been assumed. See, e.g., Tuszewski,beginning of section 1.5.2, at page 2072.

SUMMARY OF THE INVENTION

To address the problems faced by previous plasma containment systems, asystem and apparatus for containing plasma are herein described in whichplasma ions are contained magnetically in stable, large orbits andelectrons are contained electrostatically in an energy well. A majorinnovation of the present invention over all previous work with FRCs isthe simultaneous electrostatic confinement of electrons and magneticconfinement of ions, which tends to avoid anomalous transport andfacilitate classical containment of both electrons and ions. In thisconfiguration, ions may have adequate density and temperature so thatupon collisions they are fused together by the nuclear force, thusreleasing fusion energy.

In a preferred embodiment, a plasma confinement system comprises achamber, a magnetic field generator for applying a magnetic field in adirection substantially along a principle axis, and an annular plasmalayer that comprises a circulating beam of ions. Ions of the annularplasma beam layer are substantially contained within the chambermagnetically in orbits and the electrons are substantially contained inan electrostatic energy well. In one aspect of one preferred embodimenta magnetic field generator comprises a current coil. Preferably, thesystem further comprises mirror coils near the ends of the chamber thatincrease the magnitude of the applied magnetic field at the ends of thechamber. The system may also comprise a beam injector for injecting aneutralized ion beam into the applied magnetic field, wherein the beamenters an orbit due to the force caused by the applied magnetic field.In another aspect of the preferred embodiments, the system forms amagnetic field having a topology of a field reversed configuration.

Also disclosed is a method of confining plasma comprising the steps ofmagnetically confining the ions in orbits within a magnetic field andelectrostatically confining the electrons in an energy well. An appliedmagnetic field may be tuned to produce and control the electrostaticfield. In one aspect of the method the field is tuned so that theaverage electron velocity is approximately zero. In another aspect, thefield is tuned so that the average electron velocity is in the samedirection as the average ion velocity. In another aspect of the method,the method forms a field reversed configuration magnetic field, in whichthe plasma is confined.

In another aspect of the preferred embodiments, an annular plasma layeris contained within a field reversed configuration magnetic field. Theplasma layer comprises positively charged ions, wherein substantiallyall of the ions are non-adiabatic, and electrons contained within anelectrostatic energy well. The plasma layer is caused to rotate and forma magnetic self-field of sufficient magnitude to cause field reversal.

In other aspects of the preferred embodiments, the plasma may compriseat least two different ion species, one or both of which may compriseadvanced fuels.

Having a non-adiabatic plasma of energetic, large-orbit ions tends toprevent the anomalous transport of ions. This can be done in a FRC,because the magnetic field vanishes (i.e., is zero) over a surfacewithin the plasma. Ions having a large orbit tend to be insensitive toshort-wavelength fluctuations that cause anomalous transport.

Magnetic confinement is ineffective for electrons because they have asmall gyroradius—due to their small mass—and are therefore sensitive toshort-wavelength fluctuations that cause anomalous transport. Therefore,the electrons are effectively confined in a deep potential well by anelectrostatic field, which tends to prevent the anomalous transport ofenergy by electrons. The electrons that escape confinement must travelfrom the high density region near the null surface to the surface of theplasma. In so doing, most of their energy is spent in ascending theenergy well. When electrons reach the plasma surface and leave withfusion product ions, they have little energy left to transport. Thestrong electrostatic field also tends to make all the ion drift orbitsrotate in the diamagnetic direction, so that they are contained. Theelectrostatic field further provides a cooling mechanism for electrons,which reduces their radiation losses.

The increased containment ability allows for the use of advanced fuelssuch as D—He³ and p—B¹¹, as well as neutronic reactants such as D—D andD—T. In the D—He³ reaction, fast neutrons are produced by secondaryreactions, but are an improvement over the D—T reaction. The p-B¹¹reaction, and the like, is preferable because it avoids the problems offast neutrons completely.

Another advantage of the advanced fuels is the direct energy conversionof energy from the fusion reaction because the fusion products aremoving charged particles, which create an electrical current. This is asignificant improvement over Tokamaks, for example, where a thermalconversion process is used to convert the kinetic energy of fastneutrons into electricity. The efficiency of a thermal conversionprocess is lower than 30%, whereas the efficiency of direct energyconversion can be as high as 90%.

Other aspects and features of the present invention will become apparentfrom consideration of the following description taken in conjunctionwith the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments are illustrated by way of example, and not by wayof limitation, in the figures of the accompanying drawings, in whichlike reference numerals refer to like components.

FIGS. 1A and 1B show, respectively, the Lorentz force acting on apositive and a negative charge.

FIGS. 2A and 2B show Larmor orbits of charged particles in a constantmagnetic field.

FIG. 3 shows the {right arrow over (E)}×{right arrow over (B)} drift.

FIG. 4 shows the gradient drift.

FIG. 5 shows an adiabatic particle orbit in a Tokamak.

FIG. 6 shows a non-adiabatic particle orbit in a betatron.

FIG. 7 shows the International Thermonuclear Experimental Reactor(ITER).

FIG. 8 shows the magnetic field of a FRC.

FIGS. 9A and 9B show, respectively, the diamagnetic and thecounterdiamagnetic direction in a FRC.

FIG. 10 shows the colliding beam system.

FIG. 11 shows a betatron orbit.

FIGS. 12A and 12B show, respectively, the magnetic field and thedirection of the gradient drift in a FRC.

FIGS. 13A and 13B show, respectively, the electric field and thedirection of the {right arrow over (E)}×{right arrow over (B)} drift ina FRC.

FIGS. 14A, 14B and 14C show ion drift orbits.

FIGS. 15A and 15B show the Lorentz force at the ends of a FRC.

FIGS. 16A and 16B show the tuning of the electric field and the electricpotential in the colliding beam system.

FIG. 17 shows a Maxwell distribution.

FIGS. 18A and 18B show transitions from betatron orbits to drift orbitsdue to large-angle, ion-ion collisions.

FIG. 19 show A, B, C and D betatron orbits when small-angle,electron-ion collisions are considered.

FIGS. 20A, 20B and 20C show the reversal of the magnetic field in a FRC.

FIGS. 21A, 21B, 21C and 21D show the effects due to tuning of theexternal magnetic field B₀ in a FRC.

FIGS. 22A, 22B, 22C and 22D show iteration results for a D—T plasma.

FIGS. 23A, 23B, 23C, and 23D show iteration results for a D—He³ plasma.

FIGS. 24A, 24B, 24C and 24D show iteration results for a p-B¹¹ plasma.

FIG. 25 shows an exemplary confinement chamber.

FIG. 26 shows a neutralized ion beam as it is electrically polarizedbefore entering a confining chamber.

FIG. 27 is a head-on view of a neutralized ion beam as it contactsplasma in a confining chamber.

FIG. 28 is a side view schematic of a confining chamber according to apreferred embodiment of a start-up procedure.

FIG. 29 is a side view schematic of a confining chamber according toanother preferred embodiment of a start-up procedure.

FIG. 30 shows traces of B-dot probe indicating the formation of a FRC.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An ideal fusion reactor solves the problem of anomalous transport forboth ions and electrons. The anomalous transport of ions is avoided bymagnetic confinement in a field reversed configuration (FRC) in such away that the majority of the ions have large, non-adiabatic orbits,making them insensitive to short-wavelength fluctuations that causeanomalous transport of adiabatic ions. For electrons, the anomaloustransport of energy is avoided by tuning the externally applied magneticfield to develop a strong electric field, which confines themelectrostatically in a deep potential well. Moreover, the fusion fuelplasmas that can be used with the present confinement process andapparatus are not limited to neutronic fuels only, but alsoadvantageously include advanced fuels. (For a discussion of advancedfuels, see R. Feldbacher & M. Heindler, Nuclear Instruments and Methodsin Physics Research, A271 (1988)JJ-64 (North Holland Amsterdam).)

The solution to the problem of anomalous transport found herein makesuse of a specific magnetic field configuration, which is the FRC. Inparticular, the existence of a region in a FRC where the magnetic fieldvanishes makes it possible to have a plasma comprising a majority ofnon-adiabatic ions.

Background Theory

Before describing the system and apparatus in detail, it will be helpfulto first review a few key concepts necessary to understand the conceptscontained herein.

Lorentz Force and Particle Orbits in a Magnetic Field

A particle with electric charge q moving with velocity {right arrow over(v)} in a magnetic field {right arrow over (B)} experiences a force{right arrow over (F)}_(L) given by

$\begin{matrix}{{\overset{->}{F}}_{L} = {q{\frac{\overset{->}{v} \times \overset{->}{B}}{c}.}}} & (1)\end{matrix}$The force {right arrow over (F)}_(L) is called the Lorentz force. It, aswell as all the formulas used in the present discussion, is given in thegaussian system of units. The direction of the Lorentz force depends onthe sign of the electric charge q. The force is perpendicular to bothvelocity and magnetic field. FIG. 1A shows the Lorentz force 30 actingon a positive charge. The velocity of the particle is shown by thevector 32. The magnetic field is 34. Similarly, FIG. 1B shows theLorentz force 30 acting on a negative charge.

As explained, the Lorentz force is perpendicular to the velocity of aparticle; thus, a magnetic field is unable to exert force in thedirection of the particle's velocity. It follows from Newton's secondlaw, {right arrow over (F)}=m{right arrow over (a)}, that a magneticfield is unable to accelerate a particle in the direction of itsvelocity. A magnetic field can only bend the orbit of a particle, butthe magnitude of its velocity is not affected by a magnetic field.

FIG. 2A shows the orbit of a positively charged particle in a constantmagnetic field 34. The Lorentz force 30 in this case is constant inmagnitude, and the orbit 36 of the particle forms a circle. Thiscircular orbit 36 is called a Larmor orbit. The radius of the circularorbit 36 is called a gyroradius 38.

Usually, the velocity of a particle has a component that is parallel tothe magnetic field and a component that is perpendicular to the field.In such a case, the particle undergoes two simultaneous motions: arotation around the magnetic field line and a translation along it. Thecombination of these two motions creates a helix that follows themagnetic field line 40. This is indicated in FIG. 2B.

A particle in its Larmor orbit revolves around a magnetic field line.The number of radians traveled per unit time is the particle'sgyrofrequency, which is denoted by Ω and given by

$\begin{matrix}{{\Omega = \frac{qB}{mc}},} & (2)\end{matrix}$where m is the mass of the particle and c is the speed of light. Thegyroradius a_(L) of a charged particle is given by

$\begin{matrix}{{a_{L} = \frac{v_{\bot}}{\Omega}},} & (3)\end{matrix}$where v_(⊥) is the component of the velocity of the particleperpendicular to the magnetic field.

{right arrow over (E)}×{right arrow over (B)} Drift and Gradient Drift

Electric fields affect the orbits of charged particles, as shown in FIG.3. In FIG. 3, the magnetic field 44 points toward the reader. The orbitof a positively charged ion due to the magnetic field 44 alone would bea circle 36; the same is true for an electron 42. In the presence of anelectric field 46, however, when the ion moves in the direction of theelectric field 46, its velocity increases. As can be appreciated, theion is accelerated by the force q{right arrow over (E)}. It can furtherbe seen that, according to Eq. 3, the ion's gyroradius will increase asits velocity does.

As the ion is accelerated by the electric field 46, the magnetic field44 bends the ion's orbit. At a certain point the ion reverses directionand begins to move in a direction opposite to the electric field 46.When this happens, the ion is decelerated, and its gyroradius thereforedecreases. The ion's gyroradius thus increases and decreases inalternation, which gives rise to a sideways drift of the ion orbit 48 inthe direction 50 as shown in FIG. 3. This motion is called {right arrowover (E)}×{right arrow over (B)} drift. Similarly, electron orbits 52drift in the same direction 50.

A similar drift can be caused by a gradient of the magnetic field 44 asillustrated in FIG. 4. In FIG. 4, the magnetic field 44 points towardsthe reader. The gradient of the magnetic field is in the direction 56.The increase of the magnetic field's strength is depicted by the denseramount of dots in the figure.

From Eqs. 2 and 3, it follows that the gyroradius is inverselyproportional to the strength of the magnetic field. When an ion moves inthe direction of increasing magnetic field its gyroradius will decrease,because the Lorentz force increases, and vice versa. The ion'sgyroradius thus decreases and increases in alternation, which gives riseto a sideways drift of the ion orbit 58 in the direction 60. This motionis called gradient drift. Electron orbits 62 drift in the oppositedirection 64.

Adiabatic and Non-Adiabatic Particles

Most plasma comprises adiabatic particles. An adiabatic particle tightlyfollows the magnetic field lines and has a small gyroradius. FIG. 5shows a particle orbit 66 of an adiabatic particle that follows tightlya magnetic field line 68. The magnetic field lines 68 depicted are thoseof a Tokamak.

A non-adiabatic particle has a large gyroradius. It does not follow themagnetic field lines and is usually energetic. There exist other plasmasthat comprise non-adiabatic particles. FIG. 6 illustrates anon-adiabatic plasma for the case of a betatron. The pole pieces 70generate a magnetic field 72. As FIG. 6 illustrates, the particle orbits74 do not follow the magnetic field lines 72.

Radiation in Plasmas

A moving charged particle radiates electromagnetic waves. The powerradiated by the particle is proportional to the square of the charge.The charge of an ion is Ze, where e is the electron charge and Z is theatomic number. Therefore, for each ion there will be Z free electronsthat will radiate. The total power radiated by these Z electrons isproportional to the cube of the atomic number (Z³).

Charged Particles in a FRC

FIG. 8 shows the magnetic field of a FRC. The system has cylindricalsymmetry with respect to its axis 78. In the FRC, there are two regionsof magnetic field lines: open 80 and closed 82. The surface dividing thetwo regions is called the separatrix 84. The FRC forms a cylindricalnull surface 86 in which the magnetic field vanishes. In the centralpart 88 of the FRC the magnetic field does not change appreciably in theaxial direction. At the ends 90, the magnetic field does changeappreciably in the axial direction. The magnetic field along the centeraxis 78 reverses direction in the FRC, which gives rise to the term“Reversed” in Field Reversed Configuration (FRC).

In FIG. 9A, the magnetic field outside of the null surface 94 is in thedirection 96. The magnetic field inside the null surface is in thedirection 98. If an ion moves in the direction 100, the Lorentz force 30acting on it points towards the null surface 94. This is easilyappreciated by applying the right-hand rule. For particles moving in thedirection 102, called diamagnetic, the Lorentz force always pointstoward the null surface 94. This phenomenon gives rise to a particleorbit called betatron orbit, to be described below.

FIG. 9B shows an ion moving in the direction 104, calledcounterdiamagnetic. The Lorentz force in this case points away from thenull surface 94. This phenomenon gives rise to a type of orbit called adrift orbit, to be described below. The diamagnetic direction for ionsis counterdiamagnetic for electrons, and vice versa.

FIG. 10 shows a ring or annular layer of plasma 106 rotating in theions' diamagnetic direction 102. The ring 106 is located around the nullsurface 86. The magnetic field 108 created by the annular plasma layer106, in combination with an externally applied magnetic field 110, formsa magnetic field having the topology of a FRC (The topology is shown inFIG. 8).

The ion beam that forms the plasma layer 106 has a temperature;therefore, the velocities of the ions form a Maxwell distribution in aframe rotating at the average angular velocity of the ion beam.Collisions between ions of different velocities lead to fusionreactions. For this reason, the plasma beam layer 106 is called acolliding beam system.

FIG. 11 shows the main type of ion orbits in a colliding beam system,called a betatron orbit 112. A betatron orbit 112 can be expressed as asine wave centered on the null circle 114. As explained above, themagnetic field on the null circle 114 vanishes. The plane of the orbit112 is perpendicular to the axis 78 of the FRC. Ions in this orbit 112move in their diamagnetic direction 102 from a starting point 116. Anion in a betatron orbit has two motions: an oscillation in the radialdirection (perpendicular to the null circle 114), and a translationalong the null circle 114.

FIG. 12A is a graph of the magnetic field 118 in a FRC. The field 118 isderived using a one-dimensional equilibrium model, to be discussed belowin conjunction with the theory of the invention. The horizontal axis ofthe graph represents the distance in centimeters from the FRC axis 78.The magnetic field is in kilogauss. As the graph depicts, the magneticfield 118 vanishes at the null circle radius 120.

As shown in FIG. 12B, a particle moving near the null circle will see agradient 126 of the magnetic field pointing away from the null surface86. The magnetic field outside the null circle is 122, while themagnetic field inside the null circle is 124. The direction of thegradient drift is given by the cross product {right arrow over (B)}×∇B,where ∇B is the gradient of the magnetic field; thus, it can beappreciated by applying the right-hand rule that the direction of thegradient drift is in the counterdiamagnetic direction, whether the ionis outside or inside the null circle 128.

FIG. 13A is a graph of the electric field 130 in a FRC. The field 130 isderived using a one-dimensional equilibrium model, to be discussed belowin conjunction with the theory of the invention. The horizontal axis ofthe graph represents the distance in centimeters from the FRC axis 78.The electric field is in volts/cm. As the graph depicts, the electricfield 130 vanishes close to the null circle radius 120.

As shown if FIG. 13B, the electric field for ions is deconfining; itpoints away from the null surface 132,134. The magnetic field, asbefore, is in the directions 122,124. It can be appreciated by applyingthe right-hand rule that the direction of the {right arrow over(E)}×{right arrow over (B)} drift is in the diamagnetic direction,whether the ion is outside or inside the null surface 136.

FIGS. 14A and 14B show another type of common orbit in a FRC, called adrift orbit 138. Drift orbits 138 can be outside of the null surface, asshown in FIG. 14A, or inside it, as shown in FIG. 14B. Drift orbits 138rotate in the diamagnetic direction if the {right arrow over (E)}×{rightarrow over (B)} drift dominates or in the counterdiamagnetic directionif the gradient drift dominates. The drift orbits 138 shown in FIGS. 14Aand 14B rotate in the diamagnetic direction 102 from starting point 116.

A drift orbit, as shown in FIG. 14C, can be thought of as a small circlerolling over a relatively bigger circle. The small circle 142 spinsaround its axis in the sense 144. It also rolls over the big circle 146in the direction 102. The point 140 will trace in space a path similarto 138.

FIGS. 15A and 15B show the direction of the Lorentz force at the ends ofa FRC. In FIG. 15A, an ion is shown moving in the diamagnetic direction102 with a velocity 148 in a magnetic field 150. It can be appreciatedby applying the right-hand rule that the Lorentz force 152 tends to pushthe ion back into the region of closed field lines. In this case,therefore, the Lorentz force 152 is confining for the ions. In FIG. 15B,an ion is shown moving in the counterdiamagnetic direction with avelocity 148 in a magnetic field 150. It can be appreciated by applyingthe right-hand rule that the Lorentz force 152 tends to push the ioninto the region of open field lines. In this case, therefore, theLorentz force 152 is deconfining for the ions.

Magnetic and Electrostatic Confinement in a FRC

A plasma layer 106 (see FIG. 10) can be formed in a FRC by injectingenergetic ion beams around the null surface 86 in the diamagneticdirection 102 of ions. (A detailed discussion of different methods offorming the FRC and plasma ring follows below.) In the circulatingplasma layer 106, most of the ions have betatron orbits 112 (see FIG.11), are energetic, and are non-adiabatic; thus, they are insensitive toshort-wavelength fluctuations that cause anomalous transport.

While studying a plasma layer 106 in equilibrium conditions as describedabove, it was discovered that the conservation of momentum imposes arelation between the angular velocity of ions ω_(i) and the angularvelocity of electrons ω_(e). (The derivation of this relation is givenbelow in conjunction with the theory of the invention.) The relation is

$\begin{matrix}{{\omega_{e} = {\omega_{i}\left\lbrack {1 - \frac{\omega_{i}}{\Omega_{0}}} \right\rbrack}},{{{where}\mspace{14mu}\Omega_{0}} = {\frac{{ZeB}_{0}}{m_{i}c}.}}} & (4)\end{matrix}$In Eq. 4, Z is the ion atomic number, m, is the ion mass, e is theelectron charge, B₀ is the magnitude of the applied magnetic field, andc is the speed of light. There are three free parameters in thisrelation: the applied magnetic field B₀, the electron angular velocityω_(e), and the ion angular velocity ω_(i). If two of them are known, thethird can be determined from Eq. 4.

Because the plasma layer 106 is formed by injecting ion beams into theFRC, the angular velocity of ions ω_(i) is determined by the injectionkinetic energy of the beam W_(i), which is given by

$W_{i} = {{\frac{1}{2}m_{i}V_{i}^{2}} = {\frac{1}{2}{{m_{i\;}\left( {\omega_{i}r_{o}} \right)}^{2}.}}}$Here, V_(i)=ω_(i)r₀, where V_(i) is the injection velocity of ions,ω_(i) is the cyclotron frequency of ions, and r₀ is the radius of thenull surface 86. The kinetic energy of electrons in the beam has beenignored because the electron mass m_(e) is much smaller than the ionmass m_(i).

For a fixed injection velocity of the beam (fixed ω_(i)), the appliedmagnetic field B₀ can be tuned so that different values of ω_(e) areobtainable. As will be shown, tuning the external magnetic field B₀ alsogives rise to different values of the electrostatic field inside theplasma layer. This feature of the invention is illustrated in FIGS. 16Aand 16B. FIG. 16A shows three plots of the electric field (in volts/cm)obtained for the same injection velocity, ω_(i)=1.35×10⁷ s⁻¹, but forthree different values of the applied magnetic field B₀:

Plot Applied magnetic field (B₀) electron angular velocity (ω_(e)) 154B₀ = 2.77 kG ω_(e) = 0 156 B₀ = 5.15 kG ω_(e) = 0.625 × 10⁷ s⁻¹ 158 B₀ =15.5 kG ω_(e) = 1.11 × 10⁷ s⁻¹The values of ω_(e) in the table above were determined according to Eq.4. One can appreciate that ω_(e)>0 means that ω₀>ω_(i) in Eq. 4, so thatelectrons rotate in their counterdiamagnetic direction. FIG. 16B showsthe electric potential (in volts) for the same set of values of B₀ andω_(e). The horizontal axis, in FIGS. 16A and 16B, represents thedistance from the FRC axis 78, shown in the graph in centimeters. Theanalytic expressions of the electric field and the electric potentialare given below in conjunction with the theory of the invention. Theseexpressions depend strongly on ω_(e).

The above results can be explained on simple physical grounds. When theions rotate in the diamagnetic direction, the ions are confinedmagnetically by the Lorentz force. This was shown in FIG. 9A. Forelectrons, rotating in the same direction as the ions, the Lorentz forceis in the opposite direction, so that electrons would not be confined.The electrons leave the plasma and, as a result, a surplus of positivecharge is created. This sets up an electric field that prevents otherelectrons from leaving the plasma. The direction and the magnitude ofthis electric field, in equilibrium, is determined by the conservationof momentum. The relevant mathematical details are given below inconjunction with the theory of the invention.

The electrostatic field plays an essential role on the transport of bothelectrons and ions. Accordingly, an important aspect of this inventionis that a strong electrostatic field is created inside the plasma layer106, the magnitude of this electrostatic field is controlled by thevalue of the applied magnetic field B₀ which can be easily adjusted.

As explained, the electrostatic field is confining for electrons ifω_(e)>0. As shown in FIG. 16B, the depth of the well can be increased bytuning the applied magnetic field B₀. Except for a very narrow regionnear the null circle, the electrons always have a small gyroradius.Therefore, electrons respond to short-wavelength fluctuations with ananomalously fast diffusion rate. This diffusion, in fact, helps maintainthe potential well once the fusion reaction occurs. The fusion productions, being of much higher energy, leave the plasma. To maintain chargequasi-neutrality, the fusion products must pull electrons out of theplasma with them, mainly taking the electrons from the surface of theplasma layer. The density of electrons at the surface of the plasma isvery low, and the electrons that leave the plasma with the fusionproducts must be replaced; otherwise, the potential well woulddisappear.

FIG. 17 shows a Maxwellian distribution 162 of electrons. Only veryenergetic electrons from the tail 160 of the Maxwell distribution canreach the surface of the plasma and leave with fusion ions. The tail 160of the distribution 162 is thus continuously created byelectron-electron collisions in the region of high density near the nullsurface. The energetic electrons still have a small gyroradius, so thatanomalous diffusion permits them to reach the surface fast enough toaccommodate the departing fusion product ions. The energetic electronslose their energy ascending the potential well and leave with verylittle energy. Although the electrons can cross the magnetic fieldrapidly, due to anomalous transport, anomalous energy losses tend to beavoided because little energy is transported.

Another consequence of the potential well is a strong cooling mechanismfor electrons that is similar to evaporative cooling. For example, forwater to evaporate, it must be supplied the latent heat of vaporization.This heat is supplied by the remaining liquid water and the surroundingmedium, which then thermalize rapidly to a lower temperature faster thanthe heat transport processes can replace the energy. Similarly, forelectrons, the potential well depth is equivalent to water's latent heatof vaporization. The electrons supply the energy required to ascend thepotential well by the thermalization process that re-supplies the energyof the Maxwell tail so that the electrons can escape. The thermalizationprocess thus results in a lower electron temperature, as it is muchfaster than any heating process. Because of the mass difference betweenelectrons and protons, the energy transfer time from protons is about1800 times less than the electron thermalization time. This coolingmechanism also reduces the radiation loss of electrons. This isparticularly important for advanced fuels, where radiation losses areenhanced by fuel ions with atomic number Z>1.

The electrostatic field also affects ion transport. The majority ofparticle orbits in the plasma layer 106 are betatron orbits 112.Large-angle collisions, that is, collisions with scattering anglesbetween 90° and 180°, can change a betatron orbit to a drift orbit. Asdescribed above, the direction of rotation of the drift orbit isdetermined by a competition between the {right arrow over (E)}×{rightarrow over (B)} drift and the gradient drift. If the {right arrow over(E)}×{right arrow over (B)} drift dominates, the drift orbit rotates inthe diamagnetic direction. If the gradient drift dominates, the driftorbit rotates in the counterdiamagnetic direction. This is shown inFIGS. 18A and 18B. FIG. 18A shows a transition from a betatron orbit toa drift orbit due to a 180° collision, which occurs at the point 172.The drift orbit continues to rotate in the diamagnetic direction becausethe {right arrow over (E)}×{right arrow over (B)} drift dominates. FIG.18B shows another 180° collision, but in this case the electrostaticfield is weak and the gradient drift dominates. The drift orbit thusrotates in the counterdiamagnetic direction.

The direction of rotation of the drift orbit determines whether it isconfined or not. A particle moving in a drift orbit will also have avelocity parallel to the FRC axis. The time it takes the particle to gofrom one end of the FRC to the other, as a result of its parallelmotion, is called transit time; thus, the drift orbits reach an end ofthe FRC in a time of the order of the transit time. As shown inconnection with FIG. 15A, the Lorentz force at the ends is confiningonly for drift orbits rotating in the diamagnetic direction. After atransit time, therefore, ions in drift orbits rotating in thecounterdiamagnetic direction are lost.

This phenomenon accounts for a loss mechanism for ions, which isexpected to have existed in all FRC experiments. In fact, in theseexperiments, the ions carried half of the current and the electronscarried the other half. In these conditions the electric field insidethe plasma was negligible, and the gradient drift always dominated the{right arrow over (E)}×{right arrow over (B)} drift. Hence, all thedrift orbits produced by large-angle collisions were lost after atransit time. These experiments reported ion diffusion rates that werefaster than those predicted by classical diffusion estimates.

If there is a strong electrostatic field, the {right arrow over(E)}×{right arrow over (B)} drift dominates the gradient drift, and thedrift orbits rotate in the diamagnetic direction. This was shown abovein connection with FIG. 18A. When these orbits reach the ends of theFRC, they are reflected back into the region of closed field lines bythe Lorentz force; thus, they remain confined in the system.

The electrostatic fields in the colliding beam system may be strongenough, so that the {right arrow over (E)}×{right arrow over (B)} driftdominates the gradient drift. Thus, the electrostatic field of thesystem would avoid ion transport by eliminating this ion loss mechanism,which is similar to a loss cone in a mirror device.

Another aspect of ion diffusion can be appreciated by considering theeffect of small-angle, electron-ion collisions on betatron orbits. FIG.19A shows a betatron orbit 112; FIG. 19B shows the same orbit 112 whensmall-angle electron-ion collisions are considered 174; FIG. 19C showsthe orbit of FIG. 19B followed for a time that is longer by a factor often 176; and FIG. 19D shows the orbit of FIG. 19B followed for a timelonger by a factor of twenty 178. It can be seen that the topology ofbetatron orbits does not change due to small-angle, electron-ioncollisions; however, the amplitude of their radial oscillations growswith time. In fact, the orbits shown in FIGS. 19A to 19D fatten out withtime, which indicates classical diffusion.

Theory of the Invention

For the purpose of modeling the invention, a one-dimensional equilibriummodel for the colliding beam system is used, as shown in FIG. 10. Theresults described above were drawn from this model. This model shows howto derive equilibrium expressions for the particle densities, themagnetic field, the electric field, and the electric potential. Theequilibrium model presented herein is valid for a plasma fuel with onetype of ions (e.g., in a D-D reaction) or multiple types of ions (e.g.,D-T, D-He³, and p-B¹¹).

Vlasov-Maxwell Equations

Equilibrium solutions for the particle density and the electromagneticfields in a FRC are obtained by solving self-consistently theVlasov-Maxwell equations:

$\begin{matrix}{{\frac{\partial f_{j}}{\partial t} + {\left( {\overset{->}{v} \cdot \nabla} \right)f_{j}} + {{\frac{e_{j}}{m_{j}}\left\lbrack {\overset{->}{E} + {\frac{\overset{->}{v}}{c} \times \overset{->}{B}}} \right\rbrack} \cdot {\nabla_{v}f_{j}}}} = 0} & (5)\end{matrix}$

$\begin{matrix}{{\nabla{\times \overset{->}{E}}} = {{- \frac{1}{c}}\frac{\partial\overset{->}{B}}{\partial t}}} & (6)\end{matrix}$

$\begin{matrix}{{\nabla{\times \overset{->}{B}}} = {{\frac{4\;\pi}{c}{\sum\limits_{j}\;{e_{j}{\int{\overset{->}{v}f_{j}{\mathbb{d}\overset{->}{v}}}}}}} + {\frac{1}{c}\frac{\partial\overset{->}{E}}{\partial t}}}} & (7)\end{matrix}$

$\begin{matrix}{{\nabla{\cdot \overset{->}{E}}} = {4\;\pi{\sum\limits_{j}\;{e_{j}{\int{f_{j}{\mathbb{d}\overset{->}{v}}}}}}}} & (8)\end{matrix}$∇·{right arrow over (B)}=0,  (9)

where j=e, i and i=1, 2, . . . for electrons and each species of ions.In equilibrium, all physical quantities are independent of time (i.e.,∂/∂t=0). To solve the Vlasov-Maxwell equations, the followingassumptions and approximations are made:

(a) All the equilibrium properties are independent of axial position z(i.e., ∂/∂z=0). This corresponds to considering a plasma with aninfinite extension in the axial direction; thus, the model is valid onlyfor the central part 88 of a FRC.

(b) The system has cylindrical symmetry. Hence, all equilibriumproperties do not depend on θ (i.e., ∂/∂θ=0).

(c) The Gauss law, Eq. 8, is replaced with the quasi-neutralitycondition: Σ_(j)n_(j)e_(j)=0. By assuming infinite axial extent of theFRC and cylindrical symmetry, all the equilibrium properties will dependonly on the radial coordinate r. For this reason, the equilibrium modeldiscussed herein is called one-dimensional. With these assumptions andapproximations, the Vlasov-Maxwell equations reduce to:

$\begin{matrix}{{{\left( {\overset{\rightarrow}{v} \cdot \nabla} \right)f_{j}} + {\frac{e_{j}}{m_{j}}{\overset{\rightarrow}{E} \cdot {\nabla_{v}f_{j}}}} + {{\frac{e_{j}}{m_{j}c}\left\lbrack {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right\rbrack} \cdot {\nabla_{v}f_{j}}}} = 0} & (10)\end{matrix}$

$\begin{matrix}{{\nabla{\times \overset{\rightarrow}{B}}} = {\frac{4\pi}{c}{\sum\limits_{j}\;{e_{j}{\int{\overset{\rightarrow}{v}f_{j}{\mathbb{d}\overset{\rightarrow}{v}}}}}}}} & (11)\end{matrix}$

$\begin{matrix}{{\sum\limits_{a}\;{n_{j}e_{j}}} = 0.} & (12)\end{matrix}$

Rigid Rotor Distributions

To solve Eqs. 10 through 12, distribution functions must be chosen thatadequately describe the rotating beams of electrons and ions in a FRC. Areasonable choice for this purpose are the so-called rigid rotordistributions, which are Maxwellian distributions in a uniformlyrotating frame of reference. Rigid rotor distributions are functions ofthe constants of motion:

$\begin{matrix}{{{f_{j}\left( {r,\overset{\rightarrow}{v}} \right)} = {\left( \frac{m_{j}}{2\pi\; T_{j}} \right)^{\frac{3}{2}}{n_{j}(0)}{\exp\left\lbrack {- \frac{ɛ_{j} - {\omega_{j}P_{j}}}{T_{j}}} \right\rbrack}}},} & (13)\end{matrix}$where m_(j) is particle mass, {right arrow over (v)} is velocity, T_(j)is temperature, n_(j)(0) is density at r=0, and ω_(j) is a constant. Theconstants of the motion are

$ɛ_{j} = {{\frac{m_{j}}{2}v^{2}} + {e_{j}\Phi}}$(for energy) and

$P_{j} = {{m_{j}\left( {{xv}_{y} - {yv}_{x}} \right)} + {\frac{e_{j}}{c}\Psi}}$(for canonical angular momentum),where Φ is the electrostatic potential and Ψ is the flux function. Theelectromagnetic fields are

$E_{r} = {- \frac{\partial\Phi}{\partial r}}$(electric field) and

$B_{z} = {\frac{1}{r}\frac{\partial\Psi}{\partial r}}$(magnetic field).Substituting the expressions for energy and canonical angular momentuminto Eq. 13 yields

$\begin{matrix}{{{f_{j}\left( {r,\overset{\rightarrow}{v}} \right)} = {\left( \frac{m_{j}}{2\pi\; T_{j}} \right)^{\frac{3}{2}}{n_{j}(r)}\exp\left\{ {{- \frac{m_{j}}{2T_{j}}}{{\overset{\rightarrow}{v} - {{\overset{\rightarrow}{\omega}}_{j} \times \overset{\rightarrow}{r}}}}^{2}} \right\}}},} & (14)\end{matrix}$where|{right arrow over (v)}−{right arrow over (ω)} _(j) ×{right arrow over(r)}| ²=(v _(x) yω _(j))²+(v _(y) −xω _(j))² +v _(z) ²and

$\begin{matrix}{{n_{j}(r)} = {{n_{j}(0)}\exp{\left\{ {- {\frac{1}{T_{j}}\left\lbrack {{e_{j}\left( {\Phi - {\frac{\omega_{j}}{c}\Psi}} \right)} - {\frac{m_{j}}{2}\omega_{j}^{2}r^{2}}} \right\rbrack}} \right\}.}}} & (15)\end{matrix}$That the mean velocity in Eq. 14 is a uniformly rotating vector givesrise to the name rigid rotor. One of skill in the art can appreciatethat the choice of rigid rotor distributions for describing electronsand ions in a FRC is justified because the only solutions that satisfyVlasov's equation (Eq. 10) are rigid rotor distributions (e.g., Eq. 14).A proof of this assertion follows:

Proof

We require that the solution of Vlasov's equation (Eq. 10) be in theform of a drifted Maxwellian:

$\begin{matrix}{{{f_{j}\left( {\overset{\rightarrow}{r},\overset{\rightarrow}{v}} \right)} = {\left( \frac{m_{j}}{2\pi\;{T_{j}(r)}} \right)^{\frac{3}{2}}{n_{j}(r)}{\exp\left\lbrack {{- \frac{m_{\alpha}}{2{T_{j}(r)}}}\left( {\overset{\rightarrow}{v} - {{\overset{\rightarrow}{u}}_{j}(r)}} \right)^{2}} \right\rbrack}}},} & (16)\end{matrix}$i.e., a Maxwellian with particle density n_(j)(r), temperature T_(j)(r),and mean velocity u_(j)(r) that are arbitrary functions of position.Substituting Eq. 16 into the Vlasov's equation (Eq. 10) shows that (a)the temperatures T_(j)(r) must be constants; (b) the mean velocities{right arrow over (u)}_(j)(r) must be uniformly rotating vectors; and(c) the particle densities n_(j)(r) must be of the form of Eq. 15.Substituting Eq. 16 into Eq. 10 yields a third-order polynomial equationin {right arrow over (v)}:

${{\overset{->}{v} \cdot {\nabla\left( {\ln\; n_{j}} \right)}} + {{\frac{m_{j}\left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}{T_{j}} \cdot \left( {\overset{\rightarrow}{v} \cdot \nabla} \right)}{\overset{\rightarrow}{u}}_{j}} + {\frac{{m_{j}\left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}^{2}}{2T_{j}^{2}}\left( {\overset{\rightarrow}{v} \cdot \nabla} \right)T_{j}\mspace{20mu}\ldots} + {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{E} \cdot \left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}} - {{\frac{e_{j}}{T_{j}c}\left\lbrack {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right\rbrack} \cdot \left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}} = 0.$Grouping terms of like order in {right arrow over (v)} yields

${{\frac{m_{j}}{2T_{j}^{2}}{\overset{\rightarrow}{v}}^{2}\left( {\overset{\rightarrow}{v} \cdot {\nabla T_{j}}} \right)\mspace{20mu}\ldots} + {\frac{m_{j}}{T_{j}}\left( {\overset{\rightarrow}{v} \cdot {\nabla{\overset{\rightarrow}{u}}_{j}} \cdot \overset{\rightarrow}{v}} \right)} - {\frac{m_{j}}{T_{j}^{2}}\left( {\overset{\rightarrow}{v} \cdot {\overset{\rightarrow}{u}}_{j}} \right)\left( {\overset{\rightarrow}{v} \cdot {\nabla T_{j}}} \right)\mspace{14mu}\ldots} + {\overset{\rightarrow}{v} \cdot {\nabla\left( {\ln\; n_{j}} \right)}} + {\frac{m_{j}}{2T_{j}^{2}}{{\overset{\rightarrow}{u}}_{j}}^{2}\left( {\overset{\rightarrow}{v} \cdot {\nabla T_{j}}} \right)} - {\frac{m_{j}}{T_{j}}\left( {\overset{\rightarrow}{v} \cdot {\nabla{\overset{->\;}{u}}_{j}} \cdot {\overset{->}{u}}_{j}} \right)} - {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{E}}} + {\frac{e_{j}}{{cT}_{j}}{\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right) \cdot {\overset{\rightarrow}{u}}_{j}}\mspace{14mu}\ldots} + {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{E} \cdot {\overset{\rightarrow}{u}}_{j}}}} = 0.$For this polynomial equation to hold for all {right arrow over (v)}, thecoefficient of each power of {right arrow over (v)} must vanish.

The third-order equation yields T_(j)(r)=constant.

The second-order equation gives

${\overset{\rightarrow}{v} \cdot {\nabla{\overset{\rightarrow}{u}}_{j}} \cdot \overset{\rightarrow}{v}} = {{\left( {v_{x}v_{y}v_{z}} \right)\begin{pmatrix}\frac{\partial u_{x}}{\partial x} & \frac{\partial u_{y}}{\partial x} & \frac{\partial u_{z}}{\partial x} \\\frac{\partial u_{x}}{\partial y} & \frac{\partial u_{y}}{\partial y} & \frac{\partial u_{z}}{\partial y} \\\frac{\partial u_{x}}{\partial z} & \frac{\partial u_{y}}{\partial z} & \frac{\partial u_{z}}{\partial z}\end{pmatrix}\begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix}} = {{{v_{x}^{2}\frac{\partial u_{x}}{\partial x}} + {v_{y}^{2}\frac{\partial u_{y}}{\partial y}} + {v_{z}^{2}\frac{\partial u_{z}}{\partial z}} + {v_{x}{v_{y}\left( {\frac{\partial u_{y}}{\partial x} + \frac{\partial u_{x}}{\partial y}} \right)}\mspace{14mu}\ldots} + {v_{x}{v_{z}\left( {\frac{\partial u_{z}}{\partial x} + \frac{\partial u_{x}}{\partial z}} \right)}} + {v_{y}{v_{z}\left( {\frac{\partial u_{z}}{\partial y} + \frac{\partial u_{y}}{\partial z}} \right)}}} = 0.}}$For this to hold for all {right arrow over (v)}, we must satisfy

${\frac{\partial u_{x}}{\partial x} = {\frac{\partial u_{y}}{\partial y} = {\frac{\partial u_{z}}{\partial z} = {{0\mspace{14mu}{and}\mspace{14mu}\left( {\frac{\partial u_{y}}{\partial x} + \frac{\partial u_{x}}{\partial y}} \right)} = {\left( {\frac{\partial u_{z}}{\partial x} + \frac{\partial u_{x}}{\partial z}} \right) = {\left( {\frac{\partial u_{z}}{\partial y} + \frac{\partial u_{y}}{\partial z}} \right) = 0}}}}}},$which is solved generally by{right arrow over (u)} _(j)({right arrow over (r)})=({right arrow over(ω)}_(j) ×{right arrow over (r)})+{right arrow over (u)} _(0j)  (17)In cylindrical coordinates, take {right arrow over (u)}_(0j)=0 and{right arrow over (ω)}_(j)=ω_(j){circumflex over (z)}, which correspondsto injection perpendicular to a magnetic field in the {circumflex over(z)} direction. Then, {right arrow over (u)}_(j)({right arrow over(r)})=ω_(j)r{circumflex over (θ)}.

The zero order equation indicates that the electric field must be in theradial direction, i.e., {right arrow over (E)}=E_(r){circumflex over(r)}.

The first-order equation is now given by

$\begin{matrix}{{{\overset{\rightarrow}{v} \cdot {\nabla\left( {\ln\; n_{j}} \right)}} - {\frac{m_{j}}{T_{j}}\left( {\overset{\rightarrow}{v} \cdot {\nabla\;{\overset{\rightarrow}{u}}_{j}} \cdot {\overset{\rightarrow}{u}}_{j}} \right)} - {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{E}}} + {\frac{e_{j}}{{cT}_{j}}{\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right) \cdot {\overset{\rightarrow}{u}}_{j}}}} = 0.} & (18)\end{matrix}$The second term in Eq. 18 can be rewritten with

$\begin{matrix}{{{\nabla\;{\overset{\rightarrow}{u}}_{j}} \cdot {\overset{\rightarrow}{u}}_{j}} = {{\begin{pmatrix}\frac{\partial u_{r}}{\partial r} & \frac{\partial u_{\theta}}{\partial r} & \frac{\partial u_{z}}{\partial r} \\{\frac{1}{r}\frac{\partial u_{r}}{\partial\theta}} & {\frac{1}{r}\frac{\partial u_{\theta}}{\partial\theta}} & {\frac{1}{r}\frac{\partial u_{z}}{\partial\theta}} \\\frac{\partial u_{r}}{\partial z} & \frac{\partial u_{\theta}}{\partial z} & \frac{\partial u_{z}}{\partial z}\end{pmatrix}\begin{pmatrix}u_{r} \\u_{\theta} \\u_{z}\end{pmatrix}} = {{\begin{pmatrix}0 & \omega_{j} & 0 \\0 & 0 & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}0 \\{\omega_{j}r} \\0\end{pmatrix}} = {\omega_{j}^{2}r{\hat{r}.}}}}} & (19)\end{matrix}$The fourth term in Eq. 18 can be rewritten with

$\begin{matrix}{{\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right) \cdot {\overset{\rightarrow}{u}}_{j}} = {{\overset{\rightarrow}{v} \cdot \left( {\overset{\rightarrow}{B} \times {\overset{\rightarrow}{u}}_{j}} \right)} = {{\overset{\rightarrow}{v} \cdot \left( {\left( {\nabla{\times \overset{\rightarrow}{A}}} \right) \times {\overset{\rightarrow}{u}}_{j}} \right)} = {{\overset{\rightarrow}{v} \cdot \left\lbrack {\left( {\frac{1}{2}\frac{\partial}{\partial r}\left( {rA}_{\theta} \right)\hat{z}} \right) \times \left( {{- \omega_{j}}r\;\hat{\theta}} \right)} \right\rbrack} = {{\overset{\rightarrow}{v} \cdot \omega_{j}}\frac{\partial}{\partial r}\left( {rA}_{\theta} \right)\hat{r}}}}}} & (20)\end{matrix}$Using Eqs. 19 and 20, the first-order Eq. 18 becomes

${{\frac{\partial}{\partial r}\left( {\ln\; n_{j}} \right)} - {\frac{m_{j}}{T_{j}}\omega_{j}^{2}r} - {\frac{e_{j}}{T_{j}}E_{r}} + {\frac{e_{j}\omega_{j}}{{cT}_{j}}\frac{\partial}{\partial r}\left( {{rA}_{\theta}(r)} \right)}} = 0.$The solution of this equation is

$\begin{matrix}{{{n_{j}(r)} = {{n_{j}(0)}{\exp\left\lbrack {\frac{m_{j}\omega_{j}^{2}r^{2}}{2T_{j}} - \frac{e_{j}{\Phi(r)}}{T_{j}} - \frac{e_{j}\omega_{j}{{rA}_{\theta}(r)}}{{cT}_{j}}} \right\rbrack}}},} & (21)\end{matrix}$where E_(r)=−dΦ/dr and n_(j)(0) is given by

$\begin{matrix}{{n_{j}(0)} = {n_{j\; 0}{{\exp\left\lbrack {{- \frac{m_{j}\omega_{j}^{2}r_{0}^{2}}{2T_{j}}} + \frac{e_{j}{\Phi\left( r_{0} \right)}}{T_{j}} + \frac{e_{j}\omega_{j}r_{0}{A_{\theta}\left( r_{0} \right)}}{{cT}_{j}}} \right\rbrack}.}}} & (22)\end{matrix}$Here, n_(j0) is the peak density at r₀.

Solution of Vlasov-Maxwell Equations

Now that it has been proved that it is appropriate to describe ions andelectrons by rigid rotor distributions, the Vlasov's equation (Eq. 10)is replaced by its first-order moments, i.e.,

$\begin{matrix}{{{{- n_{j}}m_{j}r\;\omega_{j}^{2}} = {{n_{j}{e_{j}\left\lbrack {E_{r} + {\frac{r\;\omega_{j}}{c}B_{z}}} \right\rbrack}} - {T_{j}\frac{\mathbb{d}n_{j}}{\mathbb{d}r}}}},} & (23)\end{matrix}$which are conservation of momentum equations. The system of equations toobtain equilibrium solutions reduces to:

$\begin{matrix}\begin{matrix}{{{- n_{j}}m_{j}r\;\omega_{j}^{2}} = {{n_{j}{e_{j}\left\lbrack {E_{r} + {\frac{r\;\omega_{j}}{c}B_{z}}} \right\rbrack}} - {T_{j}\frac{\mathbb{d}n_{j}}{\mathbb{d}r}}}} & {{j = e},{i = 1},2,\ldots}\end{matrix} & (24)\end{matrix}$

$\begin{matrix}{{{- \frac{\partial}{\partial r}}\frac{1}{r}\frac{\partial\Psi}{\partial r}} = {\frac{\partial B_{z}}{\partial r} = {{\frac{4\pi}{c}j_{\theta}} = {\frac{4\pi}{c}r{\sum\limits_{j}\;{n_{j}e_{j}\omega_{j}}}}}}} & (25)\end{matrix}$

$\begin{matrix}{{\sum\limits_{j}\;{n_{j}e_{j}}} \cong 0.} & (26)\end{matrix}$

Solution for Plasma with One Type of Ion

Consider first the case of one type of ion fully stripped. The electriccharges are given by e_(j)=−e,Ze. Solving Eq. 24 for E_(r) with theelectron equation yields

$\begin{matrix}{{E_{r} = {{\frac{m}{e}r\;\omega_{e}^{2}} - {\frac{r\;\omega_{e}}{c}B_{z}} - {\frac{T_{e}}{{en}_{e}}\frac{\mathbb{d}n_{e}}{\mathbb{d}r}}}},} & (27)\end{matrix}$and eliminating E_(r) from the ion equation yields

$\begin{matrix}{{\frac{1}{r}\frac{{\mathbb{d}\;\log}\; n_{i}}{\mathbb{d}r}} = {{\frac{Z_{i}e}{c}\frac{\left( {\omega_{i} - \omega_{e}} \right)}{T_{i}}B_{z}} - {\frac{Z_{z}T_{e}}{T_{i}}\frac{1}{r}\frac{{\mathbb{d}\;\log}\; n_{e}}{\mathbb{d}r}} + \frac{m_{i}\omega_{i}^{2}}{T_{i}} + {\frac{{mZ}_{i}\omega_{e}^{2}}{T_{i}}.}}} & (28)\end{matrix}$Differentiating Eq. 28 with respect to r and substituting Eq. 25 fordB_(z)/dr yields

${{- \frac{\mathbb{d}\; B_{z}}{\mathbb{d}r}} = {{\frac{{4\pi}\;}{c}n_{e}{{er}\left( {\omega_{i} - \omega_{e}} \right)}\mspace{14mu}{and}\mspace{14mu} Z_{i}n_{i}} = n_{e}}},$with T_(e)=T_(i)=constant, and ω_(i), ω_(e), constants, obtaining

$\begin{matrix}{{\frac{1}{r}\frac{\mathbb{d}}{\mathbb{d}r}\frac{1}{r}\frac{{\mathbb{d}\;\log}\; n_{i}}{\mathbb{d}r}} = {{\frac{4\pi\; n_{e}Z_{i}e^{2}}{T_{i}}\frac{\left( {\omega_{i} - \omega_{e}} \right)^{2}}{c^{2}}} - {\frac{Z_{i}T_{e}}{T_{i}}\frac{1}{r}\frac{\mathbb{d}}{\mathbb{d}r}\frac{1}{r}{\frac{{\mathbb{d}\;\log}\; n_{e}}{\mathbb{d}r}.}}}} & (29)\end{matrix}$

The new variable ξ is introduced:

$\begin{matrix}{\xi = {\left. \frac{r^{2}}{2r_{0}^{2}}\Rightarrow{\frac{1}{r}\frac{\mathbb{d}}{\mathbb{d}r}\frac{1}{r}\frac{\mathbb{d}}{\mathbb{d}r}} \right. = {\frac{1}{r_{0}^{4}}{\frac{\mathbb{d}^{2}}{\mathbb{d}^{2}\xi}.}}}} & (30)\end{matrix}$Eq. 29 can be expressed in terms of the new variable ξ:

$\begin{matrix}{\frac{{\mathbb{d}^{2}\log}\; n_{i}}{\mathbb{d}^{2}\xi} = {{{- \frac{4\pi\; n_{e}Z_{i}e^{2}r_{0}^{4}}{T_{i}}}\frac{\left( {\omega_{i} - \omega_{e}} \right)^{2}}{c^{2}}} - {\frac{Z_{i}T_{e}}{T_{i}}{\frac{{\mathbb{d}^{2}\log}\; n_{e}}{\mathbb{d}^{2}\xi}.}}}} & (31)\end{matrix}$

Using the quasi-neutrality condition,

${n_{e} = {\left. {Z_{i}n_{i}}\Rightarrow\frac{{\mathbb{d}^{2}\log}\; n_{e}}{\mathbb{d}^{2}\xi} \right. = \frac{{\mathbb{d}^{2}\log}\; n_{i}}{\mathbb{d}^{2}\xi}}},$yields

$\begin{matrix}{\frac{{\mathbb{d}^{2}\log}\; n_{i}}{\mathbb{d}^{2}\xi} = {{{- \frac{r_{0}^{4}}{\frac{\left( {T_{i} + {Z_{i}T_{e}}} \right)}{4\;\pi\; Z_{i}^{2}e^{2}}\frac{c^{2}}{\left( {\omega_{i} - \omega_{e}} \right)^{2}}}}n_{i}} = {{{- \frac{r_{0}^{4}}{\frac{\left( {T_{e} + \frac{T_{i}}{Z_{i}}} \right)}{4\;\pi\; n_{e\; 0}e^{2}}\frac{c^{2}}{\left( {\omega_{i} - \omega_{e}} \right)^{2}}}}\frac{n_{i}}{n_{i\; 0}}} = {{- 8}\left( \frac{r_{0}}{\Delta\; r} \right)^{2}{\frac{n_{i}}{n_{i\; 0}}.}}}}} & (32)\end{matrix}$

Here is defined

$\begin{matrix}{{{r_{0}\Delta\; r} \equiv {2\sqrt{2}\left\{ \frac{T_{e} + \frac{T_{i}}{Z_{i}}}{4\pi\; n_{e\; 0}e^{2}} \right\}^{\frac{1}{2}}}}{\frac{c}{{\omega_{i} - \omega_{e}}},}} & (33)\end{matrix}$where the meaning of Δr will become apparent soon. IfN_(i)=n_(i)/n_(i0), where n_(i0) is the peak density at r=r₀, Eq. 32becomes

$\begin{matrix}{\frac{{\mathbb{d}^{2}\log}\; N_{i}}{\mathbb{d}^{2}\xi} = {{- 8}\left( \frac{r_{0}}{\Delta\; r} \right)^{2}{N_{i}.}}} & (34)\end{matrix}$

Using another new variable,

${\chi = {2\frac{r_{0}}{\Delta\; r}\xi}},$yields

${\frac{\mathbb{d}^{2}N_{i}}{\mathbb{d}^{2}\chi} = {{- 2}N_{i}}},$the solution to which is

${N_{i} = \frac{1}{\cosh^{2}\left( {\chi - \chi_{0}} \right)}},$where χ₀=χ(r₀) because of the physical requirement that N_(i)(r₀)=1.

Finally, the ion density is given by

$\begin{matrix}{n_{i} = {\frac{n_{i\; 0}}{\cosh^{2}2\left( \frac{r_{0}}{\Delta\; r} \right)\left( {\xi - \frac{1}{2}} \right)} = {\frac{n_{i\; 0}}{\cosh^{2}\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta\; r} \right)}.}}} & (35)\end{matrix}$The significance of r₀ is that it is the location of peak density. Notethat n_(i)(0)=n_(i)(√{square root over (2)}r₀). With the ion densityknown, B_(z) can be calculated using Eq. 11, and E_(r) can be calculatedusing Eq. 27.

The electric and magnetic potentials are

Φ = −∫_(r^(′) = 0)^(r′ = r)E_(r) (r^(′))𝕕r^(′)and

$\begin{matrix}{A_{\theta} = {{\frac{1}{r}{\int_{r^{\prime} = 0}^{{r'} - r}{r^{\prime}{B_{z}\left( r^{\prime} \right)}{\mathbb{d}r^{\prime}}\mspace{14mu}\Psi}}} = {{rA}_{\theta}\left( {{flux}\mspace{14mu}{function}} \right)}}} & (36)\end{matrix}$Taking r=√{square root over (2)}r₀ to be the radius at the wall (achoice that will become evident when the expression for the electricpotential Φ(r) is derived, showing that at r=√{square root over (2)}r₀the potential is zero, i.e., a conducting wall at ground), the linedensity is

$\begin{matrix}{N_{e} = {{Z_{i}N_{i}} = {{\int_{r = 0}^{r = {\sqrt{2}r_{0}}}\frac{n_{e\; 0}2\pi\; r{\mathbb{d}r}}{\cosh^{2}\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta\; r} \right)}}\  = {{2\pi\; n_{e\; 0}r_{0}\Delta\; r\mspace{14mu}\tanh\frac{r_{0}}{\Delta\; r}\mspace{14mu}\ldots} \cong {2\pi\; n_{e\; 0}r_{0}\Delta\; r\mspace{14mu}\left( {{{because}\mspace{14mu} r_{0}}\operatorname{>>}{\Delta\; r}} \right)}}}}} & (37)\end{matrix}$Thus, Δr represents an “effective thickness.” In other words, for thepurpose of line density, the plasma can be thought of as concentrated atthe null circle in a ring of thickness Δr with constant density n_(e0).

The magnetic field is

$\begin{matrix}{{B_{z}(r)} = {{B_{z}(0)} - {\frac{4\;\pi}{c}\;{\int_{r^{\prime} = 0}^{r^{\prime} = r}\mspace{7mu}{{\mathbb{d}r^{\prime}}\; n_{e}\; e\; r^{\prime}\;{\left( {\omega_{i} - \omega_{e}} \right).}}}}}} & (38)\end{matrix}$The current due to the ion and electron beams is

$\begin{matrix}{{I_{\theta} = {{\int_{0}^{\sqrt{2}r_{0}}{j_{\theta}\ {\mathbb{d}r}}} = \frac{N_{e}\; e\;\left( {\omega_{i} - \omega_{e}} \right)}{2\;\pi}}}{j_{\theta} = {n_{0}\;{er}\;{\left( {\omega_{i} - \omega_{e}} \right).}}}} & (39)\end{matrix}$Using Eq. 39, the magnetic field can be written as

$\begin{matrix}{{B_{z}(r)} = {{{B_{z}(0)} - {\frac{2\;\pi}{c}\; I_{\theta}} - {\frac{2\;\pi}{c}\; I_{\theta}\;\tan\; h\;\frac{r^{2} - r_{0}^{2}}{r_{0}\;\Delta\; r}}} = {{- B_{0}} - {\frac{2\;\pi}{c}\; I_{\theta}\;\tan\; h\;{\frac{r^{2} - r_{0}^{2}}{r_{0}\;\Delta\; r}.}}}}} & (40)\end{matrix}$In Eq. 40,

${B_{z}(0)} = {{- B_{0}} + {\frac{2\;\pi}{c}I_{\theta}}}$and

${B_{z}\left( {\sqrt{2}r_{0}} \right)} = {{- B_{0}} - {\frac{2\;\pi}{c}{I_{\theta}.}}}$If the plasma current I_(θ) vanishes, the magnetic field is constant, asexpected.

These relations are illustrated in FIGS. 20A through 20C. FIG. 20A showsthe external magnetic field {right arrow over (B)}₀ 180. FIG. 20B showsthe magnetic field due to the ring of current 182, the magnetic fieldhaving a magnitude of (2π/c)I_(θ). FIG. 20C shows field reversal 184 dueto the overlapping of the two magnetic fields 180,182.

The magnetic field is

$\begin{matrix}{{{B_{z}(r)} = {{- {B_{0}\left\lbrack {1 + {\frac{2\;\pi\; I_{\theta}}{{cB}_{0}}\tan\; h\frac{r^{2} - r_{0}^{2}}{r_{0}\Delta\; r}}} \right\rbrack}} = {- {B_{0}\left\lbrack {1 + {\sqrt{\beta}\;\tan\;{h\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta\; r} \right)}}} \right\rbrack}}}},} & (41)\end{matrix}$using the following definition for β:

$\begin{matrix}\begin{matrix}{{\frac{2\;\pi}{c}\frac{I_{\theta}}{B_{0}}} = {\frac{N_{e}{e\left( {\omega_{i} - \omega_{e}} \right)}}{{cB}_{0}} = {\frac{2\;\pi}{c}n_{e\; 0}\frac{r_{0}\Delta\;{{re}\left( {\omega_{i} - \omega_{e}} \right)}}{B_{0}}\ldots}}} \\{= {\frac{2\;\pi}{c}2{\sqrt{2}\left\lbrack \frac{T_{e} + \left( {T_{i}/Z_{i}} \right)}{4\pi\; n_{e\; 0}e^{2}} \right\rbrack}^{\frac{1}{2}}\frac{{cn}_{e\; 0}}{\omega_{i} - \omega_{e}}\frac{e\left( {\omega_{i} - \omega_{e}} \right)}{B_{0}}\ldots}} \\{= {\left\lbrack \frac{8\;{\pi\left( {{n_{e\; 0}T_{e}} + {n_{i\; 0}T_{i}}} \right)}}{B_{0}^{2}} \right\rbrack^{\frac{1}{2}} \equiv {\sqrt{\beta}.}}}\end{matrix} & (42)\end{matrix}$

With an expression for the magnetic field, the electric potential andthe magnetic flux can be calculated. From Eq. 27,

$\begin{matrix}{E_{r} = {{{{- \frac{r\;\omega_{e}}{c}}B_{z}} - {\frac{T_{e}}{e}\frac{{\mathbb{d}\ln}\; n_{e}}{\mathbb{d}r}} + {\frac{m}{e}r\;\omega_{e}^{2}}} = {- \frac{\mathbb{d}\Phi}{\mathbb{d}r}}}} & (43)\end{matrix}$

Integrating both sides of Eq. 28 with respect to r and using thedefinitions of electric potential and flux function,

$\begin{matrix}{{\Phi \equiv {- {\int_{r^{\prime} = 0}^{r^{\prime} = r}{E_{r}\ {\mathbb{d}r^{\prime}}\mspace{14mu}{and}\mspace{14mu}\Psi}}} \equiv {\int_{r^{\prime} = 0}^{r^{\prime} = r}{{B_{z}\left( r^{\prime} \right)}r^{\prime}\ {\mathbb{d}r^{\prime}}}}},} & (44)\end{matrix}$which yields

$\begin{matrix}{\Phi = {{\frac{\omega_{e}}{e}\Psi} + {\frac{T_{e}}{e}\ln\frac{n_{e}(r)}{n_{e}(0)}} - {\frac{m}{e}{\frac{r^{2}\omega_{e}^{2}}{2}.}}}} & (45)\end{matrix}$

Now, the magnetic flux can be calculated directly from the expression ofthe magnetic field (Eq. 41):

$\begin{matrix}\begin{matrix}{\Psi = {\int_{r^{\prime} = 0}^{r^{\prime} = r}{{- {B_{0}\left\lbrack {1 + {\sqrt{\beta}\tan\; h\ \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta\; r}}} \right\rbrack}}r^{\prime}{\mathbb{d}r^{\prime}}\ldots}}} \\{= {{- \frac{B_{0}r^{2}}{2}} - {\frac{B_{0}\sqrt{\beta}}{2}r_{0}\Delta\;{r\left\lbrack {{\log\left( {\cos\; h\frac{r^{2} - r_{0}^{2}}{r_{0}\Delta\; r}} \right)} - {\log\left( {\cos\; h\frac{r_{0}}{\Delta\; r}} \right)}} \right\rbrack}\ldots}}} \\{= {{- \frac{B_{0}r^{2}}{2}} + {B_{0}\frac{\sqrt{\beta}r_{0}\Delta\; r}{4}\log{\frac{n_{e}(r)}{n_{e}(0)}.}}}}\end{matrix} & (46)\end{matrix}$Substituting Eq. 46 into Eq. 45 yields

$\begin{matrix}{\Phi = {{\frac{\omega_{e}}{c}\frac{B_{0}\sqrt{\beta}r_{0}\Delta\; r}{4}\log\frac{n_{e}(r)}{n_{e}(0)}} + {\frac{T_{e}}{e}\ln\frac{n_{e}(r)}{n_{e}(0)}} - {\frac{\omega_{e}}{c}\frac{B_{0}r^{2}}{2}} - {\frac{m}{e}{\frac{r^{2}\omega_{e}^{2}}{2}.}}}} & (47)\end{matrix}$Using the definition of β,

$\begin{matrix}\begin{matrix}{{\frac{\omega_{e}}{c}B_{0}\sqrt{\beta}r_{0}\Delta\; r} = {\frac{\omega_{e}}{c}\sqrt{8\;{\pi\left( {{n_{e\; 0}T_{e}} + {n_{i\; 0}T_{i}}} \right)}}2\frac{\left( {T_{e} + {T_{i}/2}} \right)^{\frac{1}{2}}}{\sqrt{4\;\pi\; n_{e\; 0}e^{2}}}\frac{c}{\left( {\omega_{i} - \omega_{e}} \right)}\ldots}} \\{= {4\frac{\omega_{e}}{\omega_{i} - \omega_{e}}{\frac{\left( {{n_{e\; 0}T_{e}} + {n_{i\; 0}T_{i}}} \right)}{n_{e\; 0}e}.}}}\end{matrix} & (48)\end{matrix}$Finally, using Eq. 48, the expressions for the electric potential andthe flux function become

$\begin{matrix}{{\Psi(r)} = {{- \frac{B_{0}r^{2}}{2}} + {\frac{c}{\omega_{i} - \omega_{e}}\left( \frac{{n_{e\; 0}T_{e}} + {n_{i\; 0}T_{i}}}{n_{e\; 0}e} \right)\ln\frac{n_{e}(r)}{n_{e}(0)}\mspace{14mu}{and}}}} & (49)\end{matrix}$

$\begin{matrix}{{\Phi\;(r)} = {{\left\lbrack {{\frac{\omega_{e}}{\omega_{i} - \omega_{e}}\frac{\left( {{n_{e\; 0}T_{e}} + {n_{i\; 0}T_{i}}} \right)}{n_{e\; 0}e}} + \frac{T_{e}}{e}} \right\rbrack\ln\frac{n_{e}(r)}{n_{e}(0)}} - {\frac{\omega_{e}}{c}\frac{B_{0}r^{2}}{2}} - {\frac{m}{e}{\frac{r^{2}\omega_{e}^{2}}{c}.}}}} & (50)\end{matrix}$

Relationship Between ω_(i) and ω_(e)

An expression for the electron angular velocity ω_(e) can also bederived from Eqs. 24 through 26. It is assumed that ions have an averageenergy ½m_(i)(rω_(i))², which is determined by the method of formationof the FRC. Therefore, ω_(i) is determined by the FRC formation method,and ω_(e) can be determined by Eq. 24 by combining the equations forelectrons and ions to eliminate the electric field:

$\begin{matrix}{{- \left\lbrack {{n_{e}{mr}\;\omega_{e}^{2}} + {n_{i}m_{i}r\;\omega_{i}^{2}}} \right\rbrack} = {{\frac{n_{e}{er}}{c}\left( {\omega_{i} - \omega_{e}} \right)B_{z}} - {T_{e}\frac{\mathbb{d}n_{e}}{\mathbb{d}r}} - {T_{i}{\frac{\mathbb{d}n_{i}}{\mathbb{d}r}.}}}} & (51)\end{matrix}$Eq. 25 can then be used to eliminate (ω_(i)-ω_(e)) to obtain

$\begin{matrix}{\left\lbrack {{n_{e}{mr}\;\omega_{e\;}^{2}} + {n_{i}m_{i}r\;\omega_{i}^{2}}} \right\rbrack = {\frac{\mathbb{d}}{\mathbb{d}r}{\left( {\frac{B_{z}^{2}}{8\;\pi} + {\sum\limits_{j}\;{n_{j}T_{j}}}} \right).}}} & (52)\end{matrix}$Eq. 52 can be integrated from r=0 to r_(B)=√{square root over (2)}r₀.Assuming r₀/Δr>>1, the density is very small at both boundaries andB_(z)=−B₀(1±√{square root over (β)}). Carrying out the integration shows

$\begin{matrix}{{\left\lbrack {{n_{e\; 0}m\;\omega_{e\;}^{2}} + {n_{i\; 0}m_{i}\omega_{i}^{2}}} \right\rbrack r_{0}\Delta\; r} = {{\frac{B_{0}}{2\;\pi}\left\lbrack {8\;{\pi\left( {{n_{e\; 0}T_{e}} + {n_{i\; 0}T_{i}}} \right)}} \right\rbrack}^{\frac{1}{2}}.}} & (53)\end{matrix}$Using Eq. 33 for Δr yields an equation for ω_(e):

$\begin{matrix}{{{\omega_{i}^{2} + {\frac{Zm}{m_{i}}\omega_{e}^{2}}} = {\Omega_{0}\left( {\omega_{i} - \omega_{e}} \right)}},} & (54)\end{matrix}$where

$\Omega_{0} = {\frac{{ZeB}_{0}}{m_{i}c}.}$

Some limiting cases derived from Eq. 54 are:

1. ω_(i)=0 and

${\omega_{e} = {- \frac{{eB}_{0}}{mc}}};$

2. ω_(e)=0 and ω_(i)=ω₀; and

3.

$\frac{Zm}{m_{i}}\omega_{e}^{2}{\operatorname{<<}\omega_{i}^{2}}$and

$\omega_{e} \cong {{\omega_{i}\left( {1 - \frac{\omega_{i}}{\Omega_{0}}} \right)}.}$

In the first case, the current is carried entirely by electrons movingin their diamagnetic direction (ω_(e)<0). The electrons are confinedmagnetically, and the ions are confined electrostatically by

$\begin{matrix}{E_{r} = {\frac{T_{i}}{{Zen}_{i}}\frac{\mathbb{d}n_{i}}{\mathbb{d}r}\mspace{14mu}{\begin{matrix}{\underset{\_}{<}{0\mspace{14mu}{for}\mspace{14mu} r}\underset{\_}{>}r_{0}} \\{\underset{\_}{>}{0\mspace{14mu}{for}\mspace{14mu} r}\underset{\_}{<}r_{0}}\end{matrix}.}}} & (55)\end{matrix}$

In the second case, the current is carried entirely by ions moving intheir diamagnetic direction (ω_(i)>0). If ω_(i) is specified from theion energy ½m_(i)(rω_(i))², determined in the formation process, thenω_(e)=0 and Ω₀=ω_(i) identifies the value of B₀, the externally appliedmagnetic field. The ions are magnetically confined, and electrons areelectrostatically confined by

$\begin{matrix}{E_{r} = {{- \frac{T_{e}}{{en}_{e}}}\frac{\mathbb{d}n_{e}}{\mathbb{d}r}\mspace{14mu}{\begin{matrix}{\underset{\_}{>}{0\mspace{14mu}{for}\mspace{14mu} r}\underset{\_}{>}r_{0}} \\{\underset{\_}{<}{0\mspace{14mu}{for}\mspace{14mu} r}\underset{\_}{<}r_{0}}\end{matrix}.}}} & (56)\end{matrix}$

In the third case, ω_(e)>0 and Ω₀>ω_(i). Electrons move in their counterdiamagnetic direction and reduce the current density. From Eq. 33, thewidth of the distribution n_(i)(r) is increased; however, the totalcurrent/unit length is

$\begin{matrix}{{I_{\theta} = {{\int_{r = 0}^{r_{B}}{j_{\theta}\ {\mathbb{d}r}}} = {\frac{N_{e}}{2\pi}{e\left( {\omega_{i} - \omega_{e}} \right)}}}},\mspace{14mu}{where}} & (57)\end{matrix}$

$\begin{matrix}{{N_{e}{\int_{r = 0}^{r_{B}}{2\pi\; r\ {\mathbb{d}{rn}_{e}}}}} = {2\pi\; r_{0}\Delta\;{{rn}_{e\; 0}.}}} & (58)\end{matrix}$

Here, r_(B)=√{square root over (2)}r₀ and r₀Δr ∝(ω_(i)−ω_(e))⁻¹according to Eq. 33. The electron angular velocity ω_(e) can beincreased by tuning the applied magnetic field B₀. This does not changeeither I_(θ) or the maximum magnetic field produced by the plasmacurrent, which is B₀√{square root over (β)}=(2π/c)I_(θ). However, itdoes change Δr and, significantly, the potential Φ. The maximum value ofΦ is increased, as is the electric field that confines the electrons.

Tuning the Magnetic Field

In FIGS. 21A-D, the quantities n_(e)/n_(e0) 186, B_(z)/(B₀√{square rootover (β)}) 188, Φ/Φ₀ 190, and Ψ/Ψ₀ 192 are plotted against r/r₀ 194 forvarious values of B₀. The values of potential and flux are normalized toΦ₀=20(T_(e)+T_(i))/e and Ψ₀=(c/ω_(i))Φ₀. A deuterium plasma is assumedwith the following data: n_(e0)=n_(i0)=10¹⁵ cm⁻³; r₀=40 cm;½m_(i)(r₀ω_(i))²=300 keV; and T_(e)=T_(i)=100 keV. For each of the casesillustrated in FIG. 21, ω_(i)=1.35×10⁷ s⁻¹, and ω_(e) is determined fromEq. 54 for various values of B₀:

Plot applied magnetic field (B₀) electron angular velocity (ω_(e)) 154B₀ = 2.77 kG ω_(e) = 0 156 B₀ = 5.15 kG ω_(e) = 0.625 × 10⁷ s⁻¹ 158 B₀ =15.5 kG ω_(e) = 1.11 × 10⁷ s⁻¹

The case of ω_(e)=ω_(i) and B₀=1.385 kG involves magnetic confinement ofboth electrons and ions. The potential reduces toΦ/Φ₀=m_(i)(rω_(i))²/[80(T_(e)+T_(i))], which is negligible compared tothe case ω_(e)=0. The width of the density distribution Δr is reduced bya factor of 2, and the maximum magnetic field B₀√{square root over (β)}is the same as for ω_(e)=0.

Solution for Plasmas of Multiple Types of Ions

This analysis can be carried out to include plasmas comprising multipletypes of ions. Fusion fuels of interest involve two different kinds ofions, e.g., D—T, D—He³, and H-Be¹¹. The equilibrium equations (Eqs. 24through 26) apply, except that j=e, 1, 2 denotes electrons and two typesof ions where Z₁=1 in each case and Z₂=Z=1, 2, 5 for the above fuels.The equations for electrons and two types of ions cannot be solvedexactly in terms of elementary functions. Accordingly, an iterativemethod has been developed that begins with an approximate solution.

The ions are assumed to have the same values of temperature and meanvelocity V_(i)=rω_(i). Ion-ion collisions drive the distributions towardthis state, and the momentum transfer time for the ion-ion collisions isshorter than for ion-electron collisions by a factor of an order of1000. By using an approximation, the problem with two types of ions canbe reduced to a single ion problem. The momentum conservation equationsfor ions are

$\begin{matrix}{{{- n_{1}}m_{1}r\;\omega_{1}^{2}} = {{n_{1}{e\left\lbrack {E_{r} + {\frac{r\;\omega_{1}}{c}B_{z}}} \right\rbrack}} - {T_{1}\frac{\mathbb{d}n_{1}}{\mathbb{d}r}\mspace{14mu}{and}}}} & (59)\end{matrix}$

$\begin{matrix}{{{- n_{2}}m_{2}r\;\omega_{2}^{2}} = {{n_{2}{{Ze}\left\lbrack {E_{r} + {\frac{r\;\omega_{2}}{c}B_{z}}} \right\rbrack}} - {T_{2}{\frac{\mathbb{d}n_{2}}{\mathbb{d}r}\mspace{14mu}.}}}} & (60)\end{matrix}$In the present case, T₁=T₂ and ω₁=ω₂. Adding these two equations resultsin

$\begin{matrix}{{{{- n_{i}}\left\langle m_{i} \right\rangle\omega_{i}^{2}} = {{n_{i}\left\langle Z \right\rangle{e\left\lbrack {E_{r} + {\frac{r\;\omega_{i}}{c}B_{z}}} \right\rbrack}} - {T_{i}\frac{\mathbb{d}n_{i}}{\mathbb{d}r}}}},} & (61)\end{matrix}$where n_(i)=n₁+n₂; ω_(i)=ω₁=ω₂; T_(i)=T₁=T₂; n_(i)

m_(i)

=n₁m₁+n₂m₂; and n_(i)

Z

=n₁+n₂Z.

The approximation is to assume that

m_(i)

and

Z

are constants obtained by replacing n₁(r) and n₂(r) by n₁₀ and n₂₀, themaximum values of the respective functions. The solution of this problemis now the same as the previous solution for the single ion type, exceptthat

Z

replaces Z and

m_(i)

replaces m_(i). The values of n₁ and n₂ can be obtained from n₁+n₂=n_(i)and n₁+Zn₂=n_(e)=

Z

n_(i). It can be appreciated that n₁ and n₂ have the same functionalform.

Now the correct solution can be obtained by iterating the equations:

$\begin{matrix}{\frac{{\mathbb{d}\log}\mspace{14mu} N_{1}}{\mathbb{d}\xi} = {{m_{1}r_{0}^{2}\Omega_{1}\frac{\left( {\omega_{i} - \omega_{e}} \right)}{T_{i}}\frac{B_{z}(\xi)}{B_{0}}} - {\frac{T_{e}}{T_{i}}\frac{{\mathbb{d}\log}\; N_{e}}{\mathbb{d}\xi}} + {\frac{{m_{1}\left( {\omega_{i}r_{0}} \right)}^{2}}{T_{i}}\mspace{14mu}{and}}}} & (62)\end{matrix}$

$\begin{matrix}{{\frac{{\mathbb{d}\log}\; N_{2}}{\mathbb{d}\xi} = {{m_{2}r_{0}^{2}\Omega_{2}\frac{\left( {\omega_{i} - \omega_{e}} \right)}{T_{i}}\frac{B_{z}(\xi)}{B_{0}}} - {\frac{{ZT}_{e}}{T_{i}}\frac{{\mathbb{d}\log}\; N_{e}}{\mathbb{d}\xi}} + \frac{{m_{2}\left( {\omega_{i}r_{0}} \right)}^{2}}{T_{i}}}},} & (63)\end{matrix}$where

${N_{1} = \frac{n_{1}(r)}{n_{10}}},$

${N_{2} = \frac{n_{2}(r)}{n_{20}}},$

${\xi = \frac{r^{2}}{2r_{0}^{2}}},$

${\Omega_{1} = \frac{{eB}_{0}}{m_{1}\; c}},$and

$\Omega_{2} = {\frac{{ZeB}_{0}}{m_{2}\; c}.}$

The first iteration can be obtained by substituting the approximatevalues of B_(z)(ξ) and Ne(ξ) in the right hand sides of Eqs. 62 and 63and integrating to obtain the corrected values of n₁(r), n₂(r), andB_(z)(r).

Calculations have been carried out for the data shown in Table 1, below.Numerical results for fusion fuels are shown in FIGS. 22 A-D through 24A-D wherein the quantities n₁/n₁₀ 206, Φ/Φ₀ 208, and Ψ/Ψ₀ 210 areplotted against r/r₀ 204. FIGS. 22 A-D shows the first approximation(solid lines) and the final results (dotted lines) of the iterations forD-T for the normalized density of D 196, the normalized density of T198, the normalized electric potential 200, and the normalized flux 202.FIGS. 23 A-D show the same iterations for D-He³ for the normalizeddensity of D 212, the normalized density of He³ 214, the normalizedelectric potential 216, and the normalized flux 218. FIGS. 24 A-D showthe same iterations for p-B¹¹ for the normalized density of p 220, thenormalized density of B¹¹ 222, the normalized electric potential 224,and the normalized flux 226. Convergence of the iteration is most rapidfor D-T. In all cases the first approximation is close to the finalresult.

TABLE 1 Numerical data for equilibrium calculations for different fusionfuels Quantity Units D-T D-He³ p-B¹¹ n_(e0) cm⁻³ 10¹⁵ 10¹⁵ 10¹⁵ n₁₀ cm⁻³0.5 × 10¹⁵ 1/3 × 10¹⁵ 0.5 × 10¹⁵ n₂₀ cm⁻³ 0.5 × 10¹⁵ 1/3 × 10¹⁵ 10¹⁴ v₁= v₂ $\frac{cm}{\sec}$ 0.54 × 10⁹ 0.661 × 10⁹ 0.764 × 10⁹$\frac{1}{2}m_{1}v_{1}^{2}$ keV 300 450 300 $\frac{1}{2}m_{2}v_{2}^{2}$keV 450 675 3300 ω_(i) = ω₁ = ω₂ rad/s 1.35 × 10⁷ 1.65 × 10⁷ 1.91 × 10⁷r₀ cm 40 40 40 B₀ kG 5.88 8.25 15.3 <Z_(i)> None 1 3/2 1.67 <m_(i)>m_(p) 5/2 5/2 2.67$\Omega_{0} = \frac{\left\langle Z_{i} \right\rangle{eB}_{0}}{\left\langle m_{i} \right\rangle c}$rad/s 2.35 × 10⁷ 4.95 × 10⁷ 9.55 × 10⁷$\omega_{e} = {\omega_{i}\left\lbrack {1 - \frac{\omega_{i}}{\Omega_{0}}} \right\rbrack}$rad/s 0.575 × 10⁷ 1.1 × 10⁷ 1.52 × 10⁷ T_(e) keV 96 170 82 T_(i) keV 100217 235 r₀Δr cm² 114 203 313 β None 228 187 38.3Structure of the Containment System

FIG. 25 illustrates a preferred embodiment of a containment system 300according to the present invention. The containment system 300 comprisesa chamber wall 305 that defines therein a confining chamber 310.Preferably, the chamber 310 is cylindrical in shape, with principle axis315 along the center of the chamber 310. For application of thiscontainment system 300 to a fusion reactor, it is necessary to create avacuum or near vacuum inside the chamber 310. Concentric with theprinciple axis 315 is a betatron flux coil 320, located within thechamber 310. The betatron flux coil 320 comprises an electrical currentcarrying medium adapted to direct current around a long coil, as shown,which preferably comprises parallel winding multiple separate coils, andmost preferably parallel windings of about four separate coils, to forma long coil. Persons skilled in the art will appreciate that currentthrough the betatron coil 320 will result in a magnetic field inside thebetatron coil 320, substantially in the direction of the principle axis315.

Around the outside of the chamber wall 305 is an outer coil 325. Theouter coil 325 produce a relatively constant magnetic field having fluxsubstantially parallel with principle axis 315. This magnetic field isazimuthally symmetrical. The approximation that the magnetic field dueto the outer coil 325 is constant and parallel to axis 315 is most validaway from the ends of the chamber 310. At each end of the chamber 310 isa mirror coil 330. The mirror coils 330 are adapted to produce anincreased magnetic field inside the chamber 310 at each end, thusbending the magnetic field lines inward at each end. (See FIGS. 8 and10.) As explained, this bending inward of the field lines helps tocontain the plasma 335 in a containment region within the chamber 310generally between the mirror coils 330 by pushing it away from the endswhere it can escape the containment system 300. The mirror coils 330 canbe adapted to produce an increased magnetic field at the ends by avariety of methods known in the art, including increasing the number ofwindings in the mirror coils 330, increasing the current through themirror coils 330, or overlapping the mirror coils 330 with the outercoil 325.

The outer coil 325 and mirror coils 330 are shown in FIG. 25 implementedoutside the chamber wall 305; however, they may be inside the chamber310. In cases where the chamber wall 305 is constructed of a conductivematerial such as metal, it may be advantageous to place the coils 325,330 inside the chamber wall 305 because the time that it takes for themagnetic field to diffuse through the wall 305 may be relatively largeand thus cause the system 300 to react sluggishly. Similarly, thechamber 310 may be of the shape of a hollow cylinder, the chamber wall305 forming a long, annular ring. In such a case, the betatron flux coil320 could be implemented outside of the chamber wall 305 in the centerof that annular ring. Preferably, the inner wall forming the center ofthe annular ring may comprise a non-conducting material such as glass.As will become apparent, the chamber 310 must be of sufficient size andshape to allow the circulating plasma beam or layer 335 to rotate aroundthe principle axis 315 at a given radius.

The chamber wall 305 may be formed of a material having a high magneticpermeability, such as steel. In such a case, the chamber wall 305, dueto induced countercurrents in the material, helps to keep the magneticflux from escaping the chamber 310, “compressing” it. If the chamberwall were to be made of a material having low magnetic permeability,such as plexiglass, another device for containing the magnetic fluxwould be necessary. In such a case, a series of closed-loop, flat metalrings could be provided. These rings, known in the art as fluxdelimiters, would be provided within the outer coils 325 but outside thecirculating plasma beam 335. Further, these flux delimiters could bepassive or active, wherein the active flux delimiters would be drivenwith a predetermined current to greater facilitate the containment ofmagnetic flux within the chamber 310. Alternatively, the outer coils 325themselves could serve as flux delimiters.

As explained above, a circulating plasma beam 335, comprising chargedparticles, may be contained within the chamber 310 by the Lorentz forcecaused by the magnetic field due to the outer coil 325. As such, theions in the plasma beam 335 are magnetically contained in large betatronorbits about the flux lines from the outer coil 325, which are parallelto the principle axis 315. One or more beam injection ports 340 are alsoprovided for adding plasma ions to the circulating plasma beam 335 inthe chamber 310. In a preferred embodiment, the injector ports 340 areadapted to inject an ion beam at about the same radial position from theprinciple axis 315 where the circulating plasma beam 335 is contained(i.e., around the null surface). Further, the injector ports 340 areadapted to inject ion beams 350 (See FIG. 28) tangent to and in thedirection of the betatron orbit of the contained plasma beam 335.

Also provided are one or more background plasma sources 345 forinjecting a cloud of non-energetic plasma into the chamber 310. In apreferred embodiment, the background plasma sources 345 are adapted todirect plasma 335 toward the axial center of the chamber 310. It hasbeen found that directing the plasma this way helps to better containthe plasma 335 and leads to a higher density of plasma 335 in thecontainment region within the chamber 310.

Formation of the FRC

Conventional procedures used to form a FRC primarily employ the thetapinch-field reversal procedure. In this conventional method, a biasmagnetic field is applied by external coils surrounding a neutral gasback-filled chamber. Once this has occurred, the gas is ionized and thebias magnetic field is frozen in the plasma. Next, the current in theexternal coils is rapidly reversed and the oppositely oriented magneticfield lines connect with the previously frozen lines to form the closedtopology of the FRC (see FIG. 8). This formation process is largelyempirical and there exists almost no means of controlling the formationof the FRC. The method has poor reproducibility and no tuning capabilityas a result.

In contrast, the FRC formation methods of the present invention allowfor ample control and provide a much more transparent and reproducibleprocess. In fact, the FRC formed by the methods of the present inventioncan be tuned and its shape as well as other properties can be directlyinfluenced by manipulation of the magnetic field applied by the outerfield coils 325. Formation of the FRC by methods of the presentinventions also results in the formation of the electric field andpotential well in the manner described in detail above. Moreover, thepresent methods can be easily extended to accelerate the FRC to reactorlevel parameters and high-energy fuel currents, and advantageouslyenables the classical confinement of the ions. Furthermore, thetechnique can be employed in a compact device and is very robust as wellas easy to implement—all highly desirable characteristics for reactorsystems.

In the present methods, FRC formation relates to the circulating plasmabeam 335. It can be appreciated that the circulating plasma beam 335,because it is a current, creates a poloidal magnetic field, as would anelectrical current in a circular wire. Inside the circulating plasmabeam 335, the magnetic self-field that it induces opposes the externallyapplied magnetic field due to the outer coil 325. Outside the plasmabeam 335, the magnetic self-field is in the same direction as theapplied magnetic field. When the plasma ion current is sufficientlylarge, the self-field overcomes the applied field, and the magneticfield reverses inside the circulating plasma beam 335, thereby formingthe FRC topology as shown in FIGS. 8 and 10.

The requirements for field reversal can be estimated with a simplemodel. Consider an electric current I_(P) carried by a ring of majorradius r₀ and minor radius a<<r₀. The magnetic field at the center ofthe ring normal to the ring is B_(p)=2πI_(P)/(cr_(o)). Assume that thering current I_(P)=N_(p)e(Ω₀/2π) is carried by N_(p) ions that have anangular velocity Ω₀. For a single ion circulating at radius r₀=V₀/Ω₀,Ω₀=eB₀/m_(i)c is the cyclotron frequency for an external magnetic fieldB₀. Assume V₀ is the average velocity of the beam ions. Field reversalis defined as

$\begin{matrix}{{B_{p} = {\frac{N_{p}\; e\;\Omega_{0}}{r_{0}\; c} \geq {2B_{0}}}},} & (64)\end{matrix}$which implies that N_(p)>2 r₀/α_(i), and

$\begin{matrix}{{I_{p} \geq \frac{e\; V_{0}}{\pi\;\alpha_{i}}},} & (65)\end{matrix}$where α_(i)=e²/m_(i)c²=1.57×10⁻¹⁶ cm and the ion beam energy is ½m_(i)V₀². In the one-dimensional model, the magnetic field from the plasmacurrent is B_(p)=(2π/c)i_(p), where i_(p) is current per unit of length.The field reversal requirement is i_(p)>eV₀/πr₀α_(i)=0.225 kA/cm, whereB₀=69.3 G and ½m_(i)V₀ ²=100 eV. For a model with periodic rings andB_(z) is averaged over the axial coordinate

B_(z)

=(2π/c)(I_(p)/s) (s is the ring spacing), if s=r₀, this model would havethe same average magnetic field as the one dimensional model withi_(p)=I_(p)/s.

Combined Beam/Betatron Formation Technique

A preferred method of forming a FRC within the confinement system 300described above is herein termed the combined beam/betatron technique.This approach combines low energy beams of plasma ions with betatronacceleration using the betatron flux coil 320.

The first step in this method is to inject a substantially annular cloudlayer of background plasma in the chamber 310 using the backgroundplasma sources 345. Outer coil 325 produces a magnetic field inside thechamber 310, which magnetizes the background plasma. At short intervals,low energy ion beams are injected into the chamber 310 through theinjector ports 340 substantially transverse to the externally appliedmagnetic field within the chamber 310. As explained above, the ion beamsare trapped within the chamber 310 in large betatron orbits by thismagnetic field. The ion beams may be generated by an ion accelerator,such as an accelerator comprising an ion diode and a Marx generator.(see R. B. Miller, An Introduction to the Physics of Intense ChargedParticle Beams, (1982)). As one of skill in the art can appreciate, theexternally applied magnetic field will exert a Lorentz force on theinjected ion beam as soon as it enters the chamber 310; however, it isdesired that the beam not deflect, and thus not enter a betatron orbit,until the ion beam reaches the circulating plasma beam 335. To solvethis problem, the ion beams are neutralized with electrons and directedthrough a substantially constant unidirectional magnetic field beforeentering the chamber 310. As illustrated in FIG. 26, when the ion beam350 is directed through an appropriate magnetic field, the positivelycharged ions and negatively charged electrons separate. The ion beam 350thus acquires an electric self-polarization due to the magnetic field.This magnetic field may be produced by, e.g., a permanent magnet or byan electromagnet along the path of the ion beam. When subsequentlyintroduced into the confinement chamber 310, the resultant electricfield balances the magnetic force on the beam particles, allowing theion beam to drift undeflected. FIG. 27 shows a head-on view of the ionbeam 350 as it contacts the plasma 335. As depicted, electrons from theplasma 335 travel along magnetic field lines into or out of the beam350, which thereby drains the beam's electric polarization. When thebeam is no longer electrically polarized, the beam joins the circulatingplasma beam 335 in a betatron orbit around the principle axis 315, asshown in FIG. 25.

When the plasma beam 335 travels in its betatron orbit, the moving ionscomprise a current, which in turn gives rise to a poloidal magneticself-field. To produce the FRC topology within the chamber 310, it isnecessary to increase the velocity of the plasma beam 335, thusincreasing the magnitude of the magnetic self-field that the plasma beam335 causes. When the magnetic self-field is large enough, the directionof the magnetic field at radial distances from the axis 315 within theplasma beam 335 reverses, giving rise to a FRC. (See FIGS. 8 and 10). Itcan be appreciated that, to maintain the radial distance of thecirculating plasma beam 335 in the betatron orbit, it is necessary toincrease the applied magnetic field from the outer coil 325 as theplasma beam 335 increases in velocity. A control system is thus providedfor maintaining an appropriate applied magnetic field, dictated by thecurrent through the outer coil 325. Alternatively, a second outer coilmay be used to provide the additional applied magnetic field that isrequired to maintain the radius of the plasma beam's orbit as it isaccelerated.

To increase the velocity of the circulating plasma beam 335 in itsorbit, the betatron flux coil 320 is provided. Referring to FIG. 28, itcan be appreciated that increasing a current through the betatron fluxcoil 320, by Ampere's Law, induces an azimuthal electric field, E,inside the chamber 310. The positively charged ions in the plasma beam335 are accelerated by this induced electric field, leading to fieldreversal as described above. When ion beams are added to the circulatingplasma beam 335, as described above, the plasma beam 335 depolarizes theion beams.

For field reversal, the circulating plasma beam 335 is preferablyaccelerated to a rotational energy of about 100 eV, and preferably in arange of about 75 eV to 125 eV. To reach fusion relevant conditions, thecirculating plasma beam 335 is preferably accelerated to about 200 keVand preferably to a range of about 100 keV to 3.3 MeV. In developing thenecessary expressions for the betatron acceleration, the acceleration ofsingle particles is first considered. The gyroradius of ions r=V/Ω_(i)will change because V increases and the applied magnetic field mustchange to maintain the radius of the plasma beam's orbit, r₀=V/Ω_(c)

$\begin{matrix}{{\frac{\partial r}{\partial t} = {{\frac{1}{\Omega}\left\lbrack {\frac{\partial V}{\partial t} - {\frac{V}{\Omega_{i}}\frac{\partial\Omega_{i}}{\partial t}}} \right\rbrack} = 0}},} & (66)\end{matrix}$where

$\begin{matrix}{{\frac{\partial V}{\partial t} = {{\frac{r_{0}\; e}{m_{i}\; c}\;\frac{\partial B_{c}}{\partial t}} = {\frac{{eE}_{\theta}}{m_{i}} = {{- \frac{e}{m_{i}\; c}}\;\frac{1}{2\;\pi\; r_{0}}\;\frac{\partial\Psi}{\partial t}}}}},} & (67)\end{matrix}$and Ψ is the magnetic flux:

$\begin{matrix}{{\Psi = {{\int_{0}^{r_{0}}{B_{z}\; 2\;\pi\; r\ {\mathbb{d}r}}} = {\pi\; r_{0}^{2}\;\left\langle B_{z} \right\rangle}}},} & (68)\end{matrix}$where

$\begin{matrix}{\left\langle B_{z} \right\rangle = {{{- B_{F}}\;\left( \frac{r_{a}}{r_{0}} \right)^{2}} - {{B_{c}\left\lbrack {1 - \left( \frac{r_{a}}{r_{0}} \right)^{2}} \right\rbrack}.}}} & (69)\end{matrix}$From Eq. 67, it follows that

$\begin{matrix}{{\frac{\partial\left\langle B_{z} \right\rangle}{\partial t} = {{- 2}\;\frac{\partial B_{c}}{\partial t}}},} & (70)\end{matrix}$and

B_(z)

=−2B_(c)+B₀, assuming that the initial values of B_(F) and B_(c) areboth B₀. Eq. 67 can be expressed as

$\begin{matrix}{\frac{\partial V}{\partial t} = {{- \frac{e}{2\; m_{i}\; c}}\; r_{0}\;{\frac{\partial\left\langle B_{z} \right\rangle}{\partial t}.}}} & (71)\end{matrix}$After integration from the initial to final states where ½mV₀ ²=W₀ and½mV²=W, the final values of the magnetic fields are:

$\begin{matrix}{B_{c} = {{B_{0}\sqrt{\frac{W}{W_{0}}}} = {2.19\mspace{14mu}{kG}}}} & (72)\end{matrix}$and

$\begin{matrix}{{B_{F} = {{B_{0}\left\lbrack {\sqrt{\frac{W}{W_{0}}} + {\left( \frac{r_{0}}{r_{a}} \right)^{2}\;\left( {\sqrt{\frac{W}{W_{0}}} - 1} \right)}} \right\rbrack} = {10.7\mspace{14mu}{kG}}}},} & (73)\end{matrix}$assuming B₀=69.3 G, W/W₀=1000, and r₀/r_(a)=2. This calculation appliesto a collection of ions, provided that they are all located at nearlythe same radius r₀ and the number of ions is insufficient to alter themagnetic fields.

The modifications of the basic betatron equations to accommodate thepresent problem will be based on a one-dimensional equilibrium todescribe the multi-ring plasma beam, assuming the rings have spread outalong the field lines and the z-dependence can be neglected. Theequilibrium is a self-consistent solution of the Vlasov-Maxwellequations that can be summarized as follows:

(a) The density distribution is

$\begin{matrix}{{n = \frac{n_{m}}{\cos\; h^{2}\;\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\;\Delta\; r} \right)}},} & (74)\end{matrix}$which applies to the electrons and protons (assuming quasi neutrality);r₀ is the position of the density maximum; and Δr is the width of thedistribution; and

(b) The magnetic field is

$\begin{matrix}{{B_{z} = {{- B_{c}} - {\frac{2\;\pi\; I_{p}}{c}\;\tan\; h\;\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\;\Delta\; r} \right)}}},} & (75)\end{matrix}$where B_(c) is the external field produced by the outer coil 325.Initially, B_(c)=B₀. This solution satisfies the boundary conditionsthat r=r_(a) and r=r_(b) are conductors (B_(normal)=0) andequipotentials with potential Φ=0. The boundary conditions are satisfiedif r₀ ²=(r_(a) ²+r_(b) ²)/2. r_(a)=10 cm and r₀=20 cm, so it followsthat r_(b)=26.5 cm. I_(p) is the plasma current per unit length.

The average velocities of the beam particles are V_(i)=r₀ω_(i) andV_(e)=r₀ω_(e), which are related by the equilibrium condition:

$\begin{matrix}{{\omega_{e} = {\omega_{i}\left( {1 - \frac{\omega_{i}}{\Omega_{i}}} \right)}},} & (76)\end{matrix}$where Ω_(i)=eB_(c)/(m_(i)c). Initially, it is assumed B_(c)=B₀,ω_(i)=Ω_(i), and ω_(e)=0. (In the initial equilibrium there is anelectric field such that the {right arrow over (E)}×{right arrow over(B)} and the ∇B×{right arrow over (B)} drifts cancel. Other equilibriaare possible according to the choice of B_(c).) The equilibriumequations are assumed to be valid if ω_(i) and B_(c) are slowly varyingfunctions of time, but r₀=V_(i)/Ω_(i) remains constant. The conditionfor this is the same as Eq. 66. Eq. 67 is also similar, but the fluxfunction Ψ has an additional term, i.e., Ψ=πr₀ ²

B_(z)

where

$\begin{matrix}{\left\langle B_{z} \right\rangle = {{\overset{\_}{B}}_{z} + {\frac{2\;\pi}{c}{I_{p}\left( \frac{r_{b}^{2} - r_{a}^{2}}{r_{b}^{2} + r_{a}^{2}} \right)}}}} & (77)\end{matrix}$and

$\begin{matrix}{{\overset{\_}{B}}_{z} = {{- {B_{F}\left( \frac{r_{a}}{r_{0}} \right)}^{2}} - {{B_{c}\left\lbrack {1 - \left( \frac{r_{a}}{r_{0}} \right)^{2}} \right\rbrack}.}}} & (78)\end{matrix}$The magnetic energy per unit length due to the beam current is

$\begin{matrix}{{{\int_{r_{a}}^{r_{b}}{2\;\pi\; r{\mathbb{d}{r\left( \frac{B_{z} - B_{c}}{8\;\pi} \right)}^{2}}}} = {\frac{1}{2}L_{p}I_{p}^{2}}},} & (79)\end{matrix}$from which

$\begin{matrix}{{L_{p} = {\frac{r_{b}^{2} - r_{a}^{2}}{r_{b}^{2} + r_{a}^{2}}\;\frac{2\;\pi^{2}\; r_{0}^{2}}{c^{2}}\mspace{14mu}{and}}}\;{\left\langle B_{z} \right\rangle = {{\overset{\_}{B}}_{z} + {\frac{c}{\pi\; r_{0}^{2}}\; L_{p}\;{I_{p}.}}}}} & (80)\end{matrix}$The betatron condition of Eq. 70 is thus modified so that

$\begin{matrix}{{\frac{\partial{\overset{\_}{B}}_{z}}{\partial t} = {{{- 2}\;\frac{\partial B_{c}}{\partial t}} - {\frac{L_{p}}{\pi\; r_{0}^{2}}\;\frac{\partial I_{p}}{\partial t}}}},} & (81)\end{matrix}$and Eq. 67 becomes:

$\begin{matrix}{\frac{\partial V_{i}}{\partial t} = {{\frac{e}{m_{i}}\;\frac{r_{0}}{c}\;\frac{\partial B_{c}}{\partial t}} = {{{- \frac{e}{2m_{i}\; c}}r_{0}\;\frac{\partial{\overset{\_}{B}}_{z}}{\partial t}} - {\frac{e}{m_{i}}\frac{L_{p}}{2\;\pi\; r_{0}}{\frac{\partial I_{p}}{\partial t}.}}}}} & (82)\end{matrix}$After integrating,

$\begin{matrix}{{\Delta\;{\overset{\_}{B}}_{z}} = {{- 2}{{{B_{0}\left\lbrack {1 + \frac{r_{b}^{2} - r_{a}^{2}}{r_{0}^{2}}} \right\rbrack}\left\lbrack {\sqrt{\frac{W}{W_{0}}} - 1} \right\rbrack}.}}} & (83)\end{matrix}$For W₀=100 eV and W=100 keV, Δ B _(z)=−7.49 kG. Integration of Eqs. 81and 82 determines the value of the magnetic field produced by the fieldcoil:

$\begin{matrix}{B_{c} = {{B_{0}\sqrt{\frac{W}{W_{0}}}} = {2.19\mspace{14mu}{kG}}}} & (84)\end{matrix}$and

$\begin{matrix}{B_{F} = {{B_{F\; 0} - {\left( \frac{r_{0}}{r_{a}} \right)^{2}\;\Delta\;{\overset{\_}{B}}_{z}} - {\left( \frac{r_{0}^{2} - r_{a}^{2}}{r_{a}^{2}} \right)\;\Delta\; B_{c}}} = {25\mspace{14mu}{{kG}.}}}} & (85)\end{matrix}$

If the final energy is 200 keV, B_(c)=3.13 kG and B_(F)=34.5 kG. Themagnetic energy in the flux coil would be

${\frac{B_{F}^{2}}{8\;\pi}\;\pi\; r_{F}^{2}\; l} = {172\mspace{14mu}{{kJ}.}}$The plasma current is initially 0.225 kA/cm corresponding to a magneticfield of 140 G, which increases to 10 kA/cm and a magnetic field of 6.26kG. In the above calculations, the drag due to Coulomb collisions hasbeen neglected. In the injection/trapping phase, it was equivalent to0.38 volts/cm. It decreases as the electron temperature increases duringacceleration. The inductive drag, which is included, is 4.7 volts/cm,assuming acceleration to 200 keV in 100 μs.

The betatron flux coil 320 also balances the drag from collisions andinductance. The frictional and inductive drag can be described by theequation:

$\begin{matrix}{{\frac{\partial V_{b}}{\partial t} = {{- {V_{b}\left\lbrack {\frac{1}{t_{be}} + \frac{1}{t_{bi}}} \right\rbrack}} - {\frac{e}{m_{b}}\;\frac{L}{2\;\pi\; r_{0}}\;\frac{\partial I_{b}}{\partial t}}}},} & (86)\end{matrix}$where (T_(i)/m_(i))<V_(b)<(T_(e)/m). Here, V_(b) is the beam velocity,T_(e) and T_(i) are electron and ion temperatures, I_(b) is the beam ioncurrent, and

$L = {{0.01257\;{r_{0}\left\lbrack {{\ln\;\left( \frac{8r_{0}}{a} \right)} - \frac{7}{4}} \right\rbrack}} = {0.71\mspace{14mu} µ\; H}}$is the ring inductance. Also, r₀=20 cm and a=4 cm.

The Coulomb drag is determined by

$\begin{matrix}{{t_{be} = {{\frac{3}{4}\;\sqrt{\frac{2}{\pi}}\;\left( \frac{m_{i}}{m} \right)\;\frac{T_{e}^{3\text{/}2}}{{ne}^{4}\;\ln\;\Lambda}} = {195\mspace{14mu} µ\;\sec}}}{t_{bi} = {\frac{2\sqrt{2m_{i}}\; W_{b}^{3\text{/}2}}{4\;\pi\;{ne}^{4}\;\ln\;\Lambda} = {54.8\mspace{14mu} µ\;\sec}}}} & (87)\end{matrix}$To compensate the drag, the betatron flux coil 320 must provide anelectric field of 1.9 volts/cm (0.38 volts/cm for the Coulomb drag and1.56 volts/cm for the inductive drag). The magnetic field in thebetatron flux coil 320 must increase by 78 Gauss/μs to accomplish this,in which case V_(b) will be constant. The rise time of the current to4.5 kA is 18 μs, so that the magnetic field B_(F) will increase by 1.4kG. The magnetic field energy required in the betatron flux coil 320 is

$\begin{matrix}{{\frac{B_{F}^{2}}{8\;\pi} \times \pi\; r_{F}^{2}\; l} = {394\mspace{14mu}{Joules}\mspace{14mu}{\left( {l = {115\mspace{14mu}{cm}}} \right).}}} & (88)\end{matrix}$

Betatron Formation Technique

Another preferred method of forming a FRC within the confinement system300 is herein termed the betatron formation technique. This technique isbased on driving the betatron flux coil 320. A preferred embodiment ofthis technique uses the confinement system 300 depicted in FIG. 25,except that the injection of low energy ion beams is not necessary.

As indicated, the main component in the betatron formation technique isthe betatron flux coil 320 mounted in the center and along the axis ofthe chamber 310. Due to its separate parallel windings construction, thecoil 320 exhibits very low inductance and, when coupled to an adequatepower source, has a low LC time constant, which enables rapid ramp up ofthe current in the flux coil 320.

Preferably, formation of the FRC commences by energizing the externalfield coils 325, 330. This provides an axial guide field as well asradial magnetic field components near the ends to axially confine theplasma injected into the chamber 310. Once sufficient magnetic field isestablished, the background plasma sources 345 are energized from theirown power supplies. Plasma emanating from the guns streams along theaxial guide field and spreads slightly due to its temperature. As theplasma reaches the mid-plane of the chamber 310, a continuous, axiallyextending, annular layer of cold, slowly moving plasma is established.

At this point the betatron flux coil 320 is energized. The rapidlyrising current in the coil 320 causes a fast changing axial flux in thecoil's interior. By virtue of inductive effects this rapid increase inaxial flux causes the generation of an azimuthal electric field E (seeFIG. 29), which permeates the space around the flux coil. By Maxwell'sequations, this electric field is directly proportional to the change instrength of the magnetic flux inside the coil, i.e.: a faster betatroncoil current ramp-up will lead to a stronger electric field.

The inductively created electric field couples to the charged particlesin the plasma and causes a ponderomotive force, which accelerates theparticles in the annular plasma layer. Electrons, by virtue of theirsmaller mass, are the first species to experience acceleration. Theinitial current formed by this process is, thus, primarily due toelectrons. However, sufficient acceleration time (around hundreds ofmicro-seconds) will eventually also lead to ion current. Referring toFIG. 29, this electric field accelerates the electrons and ions inopposite directions. Once both species reach their terminal velocities,current is carried about equally by ions and electrons.

As noted above, the current carried by the rotating plasma gives rise toa self magnetic field. The creation of the actual FRC topology sets inwhen the self magnetic field created by the current in the plasma layerbecomes comparable to the applied magnetic field from the external fieldcoils 325, 330. At this point magnetic reconnection occurs and the openfield lines of the initial externally produced magnetic field begin toclose and form the FRC flux surfaces (see FIGS. 8 and 10).

The base FRC established by this method exhibits modest magnetic fieldand particle energies that are typically not at reactor relevantoperating parameters. However, the inductive electric acceleration fieldwill persist, as long as the current in the betatron flux coil 320continues to increase at a rapid rate. The effect of this process isthat the energy and total magnetic field strength of the FRC continuesto grow. The extent of this process is, thus, primarily limited by theflux coil power supply, as continued delivery of current requires amassive energy storage bank. However, it is, in principal,straightforward to accelerate the system to reactor relevant conditions.

For field reversal, the circulating plasma beam 335 is preferablyaccelerated to a rotational energy of about 100 eV, and preferably in arange of about 75 eV to 125 eV. To reach fusion relevant conditions, thecirculating plasma beam 335 is preferably accelerated to about 200 keVand preferably to a range of about 100 keV to 3.3 MeV. When ion beamsare added to the circulating plasma beam 335, as described above, theplasma beam 335 depolarizes the ion beams.

EXPERIMENTS Beam Trapping and FRC Formation Experiment 1 Propagating andTrapping of a Neutralized Beam in a Magnetic Containment Vessel toCreate an FRC

Beam propagation and trapping were successfully demonstrated at thefollowing parameter levels:

-   -   Vacuum chamber dimensions: about 1 m diameter, 1.5 m length.    -   Betatron coil radius of 10 cm.    -   Plasma beam orbit radius of 20 cm.    -   Mean kinetic energy of streaming beam plasma was measured to be        about 100 eV, with a density of about 10¹³ cm⁻³, kinetic        temperature on the order of 10 eV and a pulse-length of about 20        μs.    -   Mean magnetic field produced in the trapping volume was around        100 Gauss, with a ramp-up period of 150 μs. Source: Outer coils        and betatron coils.    -   Neutralizing background plasma (substantially Hydrogen gas) was        characterized by a mean density of about 10¹³ cm⁻³, kinetic        temperature of less than 10 eV.

The beam was generated in a deflagration type plasma gun. The plasmabeam source was neutral Hydrogen gas, which was injected through theback of the gun through a special puff valve. Different geometricaldesigns of the electrode assembly were utilized in an overallcylindrical arrangement. The charging voltage was typically adjustedbetween 5 and 7.5 kV. Peak breakdown currents in the guns exceeded250,000 A. During part of the experimental runs, additional pre-ionizedplasma was provided by means of an array of small peripheral cable gunsfeeding into the central gun electrode assembly before, during or afterneutral gas injection. This provided for extended pulse lengths of above25 μs.

The emerging low energy neutralized beam was cooled by means ofstreaming through a drift tube of non-conducting material beforeentering the main vacuum chamber. The beam plasma was alsopre-magnetized while streaming through this tube by means of permanentmagnets.

The beam self-polarized while traveling through the drift tube andentering the chamber, causing the generation of a beam-internal electricfield that offset the magnetic field forces on the beam. By virtue ofthis mechanism it was possible to propagate beams as characterized abovethrough a region of magnetic field without deflection.

Upon further penetration into the chamber, the beam reached the desiredorbit location and encountered a layer of background plasma provided byan array of cable guns and other surface flashover sources. Theproximity of sufficient electron density caused the beam to loose itsself-polarization field and follow single particle like orbits,essentially trapping the beam. Faraday cup and B-dot probe measurementsconfirmed the trapping of the beam and its orbit. The beam was observedto have performed the desired circular orbit upon trapping. The beamplasma was followed along its orbit for close to ¾ of a turn. Themeasurements indicated that continued frictional and inductive lossescaused the beam particles to loose sufficient energy for them to curlinward from the desired orbit and hit the betatron coil surface ataround the ¾ turn mark. To prevent this, the losses could be compensatedby supplying additional energy to the orbiting beam by inductivelydriving the particles by means of the betatron coil.

Experiment 2 FRC Formation Utilizing the Combined Beam/BetatronFormation Technique

FRC formation was successfully demonstrated utilizing the combinedbeam/betatron formation technique. The combined beam/betatron formationtechnique was performed experimentally in a chamber 1 m in diameter and1.5 m in length using an externally applied magnetic field of up to 500G, a magnetic field from the betatron flux coil 320 of up to 5 kG, and avacuum of 1.2×10⁻⁵ torr. In the experiment, the background plasma had adensity of 10¹³ cm⁻³ and the ion beam was a neutralized Hydrogen beamhaving a density of 1.2×10¹³ cm⁻³, a velocity of 2×10⁷ cm/s, and a pulselength of around 20 μs (at half height). Field reversal was observed.

Experiment 3 FRC Formation Utilizing the Betatron Formation Technique

FRC formation utilizing the betatron formation technique wassuccessfully demonstrated at the following parameter levels:

-   -   Vacuum chamber dimensions: about 1 m diameter, 1.5 m length.    -   Betatron coil radius of 10 cm.    -   Plasma orbit radius of 20 cm.    -   Mean external magnetic field produced in the vacuum chamber was        up to 100 Gauss, with a ramp-up period of 150 μs and a mirror        ratio of 2 to 1. (Source: Outer coils and betatron coils).    -   The background plasma (substantially Hydrogen gas) was        characterized by a mean density of about 10¹³ cm⁻³, kinetic        temperature of less than 10 eV.    -   The lifetime of the configuration was limited by the total        energy stored in the experiment and generally was around 30 μs.

The experiments proceeded by first injecting a background plasma layerby two sets of coaxial cable guns mounted in a circular fashion insidethe chamber. Each collection of 8 guns was mounted on one of the twomirror coil assemblies. The guns were azimuthally spaced in anequidistant fashion and offset relative to the other set. Thisarrangement allowed for the guns to be fired simultaneously and therebycreated an annular plasma layer.

Upon establishment of this layer, the betatron flux coil was energized.Rising current in the betatron coil windings caused an increase in fluxinside the coil, which gave rise to an azimuthal electric field curlingaround the betatron coil. Quick ramp-up and high current in the betatronflux coil produced a strong electric field, which accelerated theannular plasma layer and thereby induced a sizeable current.Sufficiently strong plasma current produced a magnetic self-field thataltered the externally supplied field and caused the creation of thefield reversed configuration. Detailed measurements with B-dot loopsidentified the extent, strength and duration of the FRC.

An example of typical data is shown by the traces of B-dot probe signalsin FIG. 30. The data curve A represents the absolute strength of theaxial component of the magnetic field at the axial mid-plane (75 cm fromeither end plate) of the experimental chamber and at a radial positionof 15 cm. The data curve B represents the absolute strength of the axialcomponent of the magnetic field at the chamber axial mid-plane and at aradial position of 30 cm. The curve A data set, therefore, indicatesmagnetic field strength inside of the fuel plasma layer (betweenbetatron coil and plasma) while the curve B data set depicts themagnetic field strength outside of the fuel plasma layer. The dataclearly indicates that the inner magnetic field reverses orientation (isnegative) between about 23 and 47 μs, while the outer field stayspositive, i.e., does not reverse orientation. The time of reversal islimited by the ramp-up of current in the betatron coil. Once peakcurrent is reached in the betatron coil, the induced current in the fuelplasma layer starts to decrease and the FRC rapidly decays. Up to nowthe lifetime of the FRC is limited by the energy that can be stored inthe experiment. As with the injection and trapping experiments, thesystem can be upgraded to provide longer FRC lifetime and accelerationto reactor relevant parameters.

Overall, this technique not only produces a compact FRC, but it is alsorobust and straightforward to implement. Most importantly, the base FRCcreated by this method can be easily accelerated to any desired level ofrotational energy and magnetic field strength. This is crucial forfusion applications and classical confinement of high-energy fuel beams.

Experiment 4 FRC Formation Utilizing the Betatron Formation Technique

An attempt to form an FRC utilizing the betatron formation technique hasbeen performed experimentally in a chamber 1 m in diameter and 1.5 m inlength using an externally applied magnetic field of up to 500 G, amagnetic field from the betatron flux coil 320 of up to 5 kG, and avacuum of 5×10⁻⁶ torr. In the experiment, the background plasmacomprised substantially Hydrogen with of a density of 10¹³ cm⁻³ and alifetime of about 40 μs. Field reversal was observed.

Fusion

Significantly, these two techniques for forming a FRC inside of acontainment system 300 described above, or the like, can result inplasmas having properties suitable for causing nuclear fusion therein.More particularly, the FRC formed by these methods can be accelerated toany desired level of rotational energy and magnetic field strength. Thisis crucial for fusion applications and classical confinement ofhigh-energy fuel beams. In the confinement system 300, therefore, itbecomes possible to trap and confine high-energy plasma beams forsufficient periods of time to cause a fusion reaction therewith.

To accommodate fusion, the FRC formed by these methods is preferablyaccelerated to appropriate levels of rotational energy and magneticfield strength by betatron acceleration. Fusion, however, tends torequire a particular set of physical conditions for any reaction to takeplace. In addition, to achieve efficient burn-up of the fuel and obtaina positive energy balance, the fuel has to be kept in this statesubstantially unchanged for prolonged periods of time. This isimportant, as high kinetic temperature and/or energy characterize afusion relevant state. Creation of this state, therefore, requiressizeable input of energy, which can only be recovered if most of thefuel undergoes fusion. As a consequence, the confinement time of thefuel has to be longer than its burn time. This leads to a positiveenergy balance and consequently net energy output.

A significant advantage of the present invention is that the confinementsystem and plasma described herein are capable of long confinementtimes, i.e., confinement times that exceed fuel burn times. A typicalstate for fusion is, thus, characterized by the following physicalconditions (which tend to vary based on fuel and operating mode):

Average ion temperature: in a range of about 30 to 230 keV andpreferably in a range of about 80 keV to 230 keV

Average electron temperature: in a range of about 30 to 100 keV andpreferably in a range of about 80 to 100 keV

Coherent energy of the fuel beams (injected ion beams and circulatingplasma beam): in a range of about 100 keV to 3.3 MeV and preferably in arange of about 300 keV to 3.3 MeV.

Total magnetic field: in a range of about 47.5 to 120 kG and preferablyin a range of about 95 to 120 kG (with the externally applied field in arange of about 2.5 to 15 kG and preferably in a range of about 5 to 15kG).

Classical Confinement time: greater than the fuel burn time andpreferably in a range of about 10 to 100 seconds.

Fuel ion density: in a range of about 10¹⁴ to less than 10¹⁶ cm⁻³ andpreferably in a range of about 10¹⁴ to 10¹⁵ cm⁻³.

Total Fusion Power: preferably in a range of about 50 to 450 kW/cm(power per cm of chamber length)

To accommodate the fusion state illustrated above, the FRC is preferablyaccelerated to a level of coherent rotational energy preferably in arange of about 100 keV to 3.3 MeV, and more preferably in a range ofabout 300 keV to 3.3 MeV, and a level of magnetic field strengthpreferably in a range of about 45 to 120 kG, and more preferably in arange of about 90 to 115 kG. At these levels, high energy ion beams canbe injected into the FRC and trapped to form a plasma beam layer whereinthe plasma beam ions are magnetically confined and the plasma beamelectrons are electrostatically confined.

Preferably, the electron temperature is kept as low as practicallypossible to reduce the amount of bremsstrahlung radiation, which can,otherwise, lead to radiative energy losses. The electrostatic energywell of the present invention provides an effective means ofaccomplishing this.

The ion temperature is preferably kept at a level that provides forefficient burn-up since the fusion cross-section is a function of iontemperature. High direct energy of the fuel ion beams is essential toprovide classical transport as discussed in this application. It alsominimizes the effects of instabilities on the fuel plasma. The magneticfield is consistent with the beam rotation energy. It is partiallycreated by the plasma beam (self-field) and in turn provides the supportand force to keep the plasma beam on the desired orbit.

Fusion Products

The fusion products are born predominantly near the null surface fromwhere they emerge by diffusion towards the separatrix 84 (see FIG. 8).This is due to collisions with electrons (as collisions with ions do notchange the center of mass and therefore do not cause them to changefield lines). Because of their high kinetic energy (product ions havemuch higher energy than the fuel ions), the fusion products can readilycross the separatrix 84. Once they are beyond the separatrix 84, theycan leave along the open field lines 80 provided that they experiencescattering from ion-ion collisions. Although this collisional processdoes not lead to diffusion, it can change the direction of the ionvelocity vector such that it points parallel to the magnetic field.These open field lines 80 connect the FRC topology of the core with theuniform applied field provided outside the FRC topology. Product ionsemerge on different field lines, which they follow with a distributionof energies; advantageously in the form of a rotating annular beam. Inthe strong magnetic fields found outside the separatrix 84 (typicallyaround 100 kG), the product ions have an associated distribution ofgyro-radii that varies from a minimum value of about 1 cm to a maximumof around 3 cm for the most energetic product ions.

Initially the product ions have longitudinal as well as rotationalenergy characterized by ½M(v_(par))² and ½M(v_(perp))². v_(perp) is theazimuthal velocity associated with rotation around a field line as theorbital center. Since the field lines spread out after leaving thevicinity of the FRC topology, the rotational energy tends to decreasewhile the total energy remains constant. This is a consequence of theadiabatic invariance of the magnetic moment of the product ions. It iswell known in the art that charged particles orbiting in a magneticfield have a magnetic moment associated with their motion. In the caseof particles moving along a slow changing magnetic field, there alsoexists an adiabatic invariant of the motion described by½M(v_(perp))²/B. The product ions orbiting around their respective fieldlines have a magnetic moment and such an adiabatic invariant associatedwith their motion. Since B decreases by a factor of about 10 (indicatedby the spreading of the field lines), it follows that v_(perp) willlikewise decrease by about 3.2. Thus, by the time the product ionsarrive at the uniform field region their rotational energy would be lessthan 5% of their total energy; in other words almost all the energy isin the longitudinal component.

While the invention is susceptible to various modifications andalternative forms, a specific example thereof has been shown in thedrawings and is herein described in detail. It should be understood,however, that the invention is not to be limited to the particular formdisclosed, but to the contrary, the invention is to cover allmodifications, equivalents, and alternatives falling within the spiritand scope of the appended claims.

1. An apparatus for containing plasma comprising a generally cylindricalchamber, a means for creating an azimuthally symmetric magnetic fieldwithin the chamber with a flux substantially parallel to a principleaxis of the chamber and having open field lines, a means for creating anazimuthal electric field within the chamber positioned about andextending along the principle axis, and a means for injecting plasmacontaining electrons and ions into the chamber.
 2. The apparatus ofclaim 1 wherein the means for generating the magnetic field comprises aplurality of field coils extending about the chamber.
 3. The apparatusof claim 2 wherein means for generating the magnetic field furthercomprises a plurality of mirror coils.
 4. The apparatus of claim 1wherein the means for generating the magnetic field is tunable.
 5. Theapparatus of claim 4 further comprising a control system coupled to themeans for generating the magnetic field.
 6. The apparatus of claim 1wherein the means for creating the azimuthal electric field includes acurrent coil positioned along the principal axis of the chamber.
 7. Theapparatus of claim 6 wherein the current coil includes parallel windingsof a plurality of separate coils.
 8. The apparatus of claim 1 furthercomprising ion beam injectors coupled to the chamber to inject ion beamsinto the chamber.
 9. The apparatus of claim 1 wherein the chamber has agenerally annularly shaped cross-section.